Question 1

Let us try to find a few more pairs of numbers from their sums and differences:
(a) Sum = 27, Difference = 9
(b) Sum = 4, Difference = 12
(c) Sum = 0, Difference = 10
(d) Sum = 0, Difference = -10
(e) Sum = -7, Difference = -1
(f) Sum = -7, Difference = -13

Accepted Answer

(a) Sum = 27, Difference = 9
Hence correct pair is (18, 9).
(b) Sum = 4, Difference = 12
Hence correct pair is (8, -4).
(c) Sum = 0, Difference = 10
Hence correct pair is (5, -5).
(d) Sum = 0, Difference = -10
Hence correct pair is (8, -8).
(e) Sum = -7, Difference = -1
Hence, the correct pair is (-4, -3).
(f) Sum = -7, Difference = -13
Hence correct pair is (-10, 3).
2.2 Multiplication of Integers
Figure It Out (Page 31)

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 2

If 123 × 456 = 56088, without calculating, find the value of:
(a) (-123) × 456
(b) (-123) × (-456)
(c) (123) × (-456)

Accepted Answer

(a) Given 123 × 456 = 56088 …..(i)
Then(a) (-123) × 456
This is the product of a negative and a positive integer, so the result will be negative.
i.e. -(123) × 456 = -(123 × 456) = -56088
(b) (-123) × (-456)
This is the product of two negative integers, so the result will be positive.
∴ (-123) × (-456) = 123 × 456 = 56088
(c) 123 × (-456)
This is the product of a positive and a negative integer, so the result will be negative.
∴ 123 × (-456) = -(123 × 456) = -56088

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 3

Try to frame a simple rule to multiply two integers.

Accepted Answer

Rule for multiplying two integers:
Multiply their absolute values.
If the integers have different signs, the product is negative.
If both integers have the same sign, the product is positive.
Figure It Out (Pages 33-34)

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 4

Replace the blank with an integer to make a true statement.
(a) (-3) × ________ = 27
(b) 5 × ________ = (-35)
(c) ________ × (-8) = (-56)
(d) ________ × (-12) = 132
(e) ________ ÷ (-8) = 7
(f) ________ ÷ 12 = -11

Accepted Answer

(a) Let -3 × x = 27
⇒ x = $\frac{27}{-3}=-\frac{(27)}{3}$ = -9
(b) 5 × x = -35
⇒ x = $\frac{-35}{5}=-\frac{(35)}{5}$ = -7
(c) x × -8 = -56
⇒ x = $\frac {-56}{-8}$ = 7
(d) x × (-12) = 132
⇒ x = $\frac{132}{-12}=-\frac{-(132)}{12}$ = -11
(e) x ÷ (-8) = 7
⇒ x × $\frac {1}{-8}$ = 7
⇒ x = 7 × (-8) = -56
(f) x ÷ 12 = -11
⇒ x × $\frac {1}{12}$ = -11
⇒ x = -11 × 12 = -132
Figure It Out (Pages 42-44)

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 5

Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is — start with any number; if the number is even, take half of it; if the number is odd, multiply it by -3 and add 1; repeat. An example sequence is shown below.
Try this with different starting numbers: (-21), (-6), and so on. Describe the patterns you observe.

Accepted Answer

(a) Following the rules for the sequence starting with -7.
The rule is if even, take half, if odd, multiply by -3, and add 1.
Start with -7 (odd)
[(-7) × (-3)] + 1 = 21 + 1 = 22 (even)
then 22 ÷ 2 = 11 (odd)
then 11 × (-3) + 1 = -33 + 1 = -32 (even)
then (-32) ÷ 2 = -16 (even)
then (-16) ÷ 2 = -8 (even)
then (-8) ÷ 2 = -4 (even)
then (-4) ÷ 2 = -2 (even)
then (-2) ÷ 2 = – 1 (odd)
then [(-1) × (-3)] + 1 = 4 (even)
then 4 ÷ 2 = 2 (even)
then 2 ÷ 2 = 1 (odd)
(b) (i) Now for the starting number -21 (odd)
then [(-21) × (-3)] + 1 = 63 + 1 = 64 (even)
then 64 ÷ 2 = 32 (even)
then 32 ÷ 2 = 16 (even)
then 16 + 2 = 8 (even)
then 8 ÷ 2 = 4 (even)
then 4 ÷ 2 = 2 (even)
then 2 ÷ 2 = 1 (odd)
then 1 × (-3) + 1 = -2 (even)
then (-2) ÷ 2 = -1 (odd)
then [-1 × (-3)] + 1 = 4 (even)
then 4 ÷ 2 = 2 (even)
then 2 ÷ 2 = 1 (odd)
then 1 × (-3) + 1 = -2 (even)
Hence the sequence for -21 is
(ii) For the sequence, the starting number is -6
-6 is even then -6 + 2 = -3 (odd)
then [(-3) × (-3)] + 1 = 9 + 1 = 10 (even)
then 10 ÷ 2 = 5 (odd)
5 × (-3) + 1 = -15 + 1 = -14 (even)
then -14 ÷ 2 = -7 (odd)
[(-7) × (-3)] + 1 = +21 + 1 = 22 (even)
then 22 ÷ 2 = 11 (odd)
then [11 × (-3)] + 1 = -33 + 1 = -32 (even)
then -32 ÷ 2 = -16 (even)
then -16 ÷ 2 = -8 (even)
then -8 ÷ 2 = -4 (even)
then -4 ÷ 2 = -2 (even)
then (-2) ÷ 2 = -1 (odd)
then [(-1) × (-3)] + 1 = 3 + 1 = 4 (even)
then 4 ÷ 2 = 2 (even)
then 2 ÷ 2 = 1 (odd)
then 1 × (-3) + 1 = -2
Hence, the sequence is
Observation: For numbers like -21, -6, etc., the sequences eventually reach a repeating loop of -2, -1, 4, 2, 1, -2.
All starting numbers end up in this cycle.

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 6

In a test, (+4) marks are given for every correct answer and (-2) marks are given for every incorrect answer.
(a) Anita answered all the questions in the test. She scored 40 marks even though 15 of her answers were correct. How many of her answers were incorrect? How many questions are in the test?
(b) Anil scored (-10) marks even though he had 5 correct answers. How many of his answers were incorrect? Did he leave any questions unanswered?

Accepted Answer

(a) Anita had 15 correct answers, and each correct answer is worth +4 marks.
Marks from correct answers = 15 × 4 = 60
∴ Anita total score = 40
Marks lost = 60 – 40 = 20
Each incorrect answer is worth -2 marks
∴ No. of incorrect answers = $\frac {20}{2}$ = 10
Anita had 10 incorrect answers.
Total no. of questions = sum of correct and incorrect answers
= 15 + 10
= 25
Hence, Anita had 10 incorrect answers out of 25 questions on the test.
(b) Anil had 5 correct answers, and each correct answer is worth + 4 marks they 5 × 4 = 20 marks.
From incorrect answers = -10 – 20 = -30.
Each incorrect answer is worth -2 marks
∴ $\frac {-30}{-2}$ = 15
Hence, Anil had 15 incorrect answers.
Yes, he leaves some questions unanswered.
Anil answered the question = (Total No. of Questions) – (Anil’s correct answers + Anil’s incorrect answers)
= 25 – (5 + 15)
= 25 – 20
= 5
Hence, Anil left 5 questions unanswered.

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 7

Pick the pattern – find the operations done by the machine shown below:

Accepted Answer

Operation done by machine is (First Number) – [Second number × Third Number]
I row: 4 – [(8) × -3] = 4 – [-24] = 4 + 24 = 28.
II row: 6 – [9 × 6] = 6 – [54] = 6 – 54 = -48.
III row: 2 – [3 × (-2)] = 2 – (-6) = 2 + 6 = 8.
IV row: -9 – [5 × (-8)] = -9 – (-40) = -9 + 40 = 31.
V row: 7 – (-4 × -6) = 7 – (24) = 7 – 24 = -17.
VI row: -16 – (-6 × -9) = -16 – (54) = -70.
Thus, we have found the pattern that satisfies all the values.
Hence, the missing operation is -70.

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 8

Imagine you’re in a place where the temperature drops by 5°C each hour. If the temperature is currently at 8°C, write an expression that denotes the temperature after 4 hours.

Accepted Answer

Current temperature = 8°C
The temperature drops by 5°C each hour
Total drop in temperature = Temperature drop in 1 hour × no. of hours = 5 × 4
Temperature after 4 hours = 8 – 4 × 5
Hence required expression is 8 – (5 × 4).

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 9

Find 3 consecutive numbers with a product of (a) -6, (b) 120.

Accepted Answer

(a) Let three consecutive numbers be n – 1, n, and n + 1.
Then product = (n – 1) (n) (n + 1) = -6
The only consecutive integers whose product is -6 are -2, -1, and 0.
But -2 × -1 × o = 0, not possible.
Consecutive numbers are integers.
The product is negative, so there must be an odd number of negative integers.
Integers are -2, -1, 1.
Their product = – 2 × -1 × 1 = 2, not possible
Integers are -3, -2, -1.
Their product = [-3 × -2] × -1 = 6 × -1 = -6
Hence, consecutive integers are -3, -2, -1.
(b) Let three consecutive numbers be n – 1, n, and n + 1.
∴ (n – 1) (n) (n + 1) = 120
Now the cube root of 120 = 4.93
So n is likely to be 5.
∴ (5 – 1) × (5) × (5 + 1) = 4 × 5 × 6 = 120
Hence, consecutive integers are 4, 5, and 6.

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 10

An alien society uses a peculiar currency called ‘pibs’ with just two denominations of coins — a + 13 pibs coin and a – 9 pibs coin. You have several of these coins. Is it possible to purchase an item that costs +85 pibs? Yes, we can use 10 coins of +13 pibs and 5 coins of -9 pibs to make a total of +85. Using the two denominations, try to get the following totals:
(a) +20
(b) +40
(c) -50
(d) +8
(e) +10
(f) -2
(g) +1
[Hint: Writing down a few multiples of 13 and 9 can help.]
(h) Is it possible t

Accepted Answer

The currency has two denominations: +13 pibs and -9 pibs.
We need to determine if it is possible to make the given totals.
This is a linear equation of the form 13x – 9y = total,
where x and y are non-negative integers.
(a) Now total = +20
Then we need to find integers 13x – 9y = 20, x, y > 0 such that
If x = 1, then 13 – 9y = 20 ⇒ -9y = 7. No solution.
If x = 2, then 26 – 9y = 20 ⇒ -9y = -6. No solution.
IF x = 3 then 39 – 9y = 20 ⇒ -9y = -19. No solution.
If x = 4, then 52 – 9y = 20 ⇒ -9y = -32. No solution.
If x = 5 then 65 – 9y = 20 ⇒ -9y = -45 ⇒ y = 5
Hence, x = 5, y = 5 is a valid solution.
(b) +40
Take 10 coins of +13 and 10 coins of -9:
10 × 13 – 10 × 9 = 130 – 90 = +40 pibs.
(c) -50
Take 10 coins of +13 and 20 coins of -9:
10 × 13 – 20 × 9 = 130 – 180 = – 50 pibs.
(d) +8
Take 2 coins of +13 and 2 coins of -9:
2 × 13 – 2 × 9 = 26 – 18 = +8 pibs.
(e) +10
Take 7 coins of +13 and 9 coins of -9:
7 × 13 – 9 × 9 = 91 – 81 = +10 pibs.
(f) -2
Take 13 coins of +13 and 19 coins of -9:
13 × 13 – 19 × 9 = 169 – 171 = -2 pibs.
(g) +1
Take 7 coins of +13 and 10 coins of -9:
7 × 13 – 10 × 9 = 91 – 90 = +1 pibs.
(h) Yes, it is possible.
Take 122 coins of +13 and 2 coins of -9:
122 × 13 – 2 × 9 = 1586 – 18 = 1568 pibs.

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 11

Find the values of:
(a) (32 × (-18)) ÷ ((-36))
(b) (32) ÷ ((-36) × (-18))
(c) (25 × (-12)) ÷ ((45) × (-27))
(d) (280 × (-7)) ÷ ((-8) × (-35))

Accepted Answer

(a) Here 32 × (-18) ÷ (-36)
= -576 ÷ (-36)
= 16
(b) Here 32 ÷ [(-36) × (-18)]
= (32) ÷ 648
= $\frac {4}{81}$
(c) Here [25 × (-12)] ÷ [45 × (-27)]
= (-300) ÷ (-1215)
= $\frac {20}{81}$
(d) Here [280 × (-7)] ÷ [(-8) × (-35)]
= (-1960) ÷ 280
= -7

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 12

Arrange the expressions given below in increasing order:
(a) (-348) + (-1064)
(b) (-348) – (-1064)
(c) 348 – (-1064)
(d) (-348) × (-1064)
(e) 348 × (-1064)
(f) 348 × 964

Accepted Answer

(a) (-348) + (-1064)
Add two negative numbers → keep negative, add absolute values:
-348 + (-1064) = -1412.
(b) (-348) – (-1064)
Subtracting a negative → same as adding positive:
-348 – (-1064) = -348 + 1064 = 716.
(c) 348 – (-1064)
Subtracting a negative → add positive.
348 – (-1064) = 348 + 1064 = 1412.
(d) (-348) × (-1064)
Negative × Negative = Positive
(-348) × (-1064) = 370272
(e) 348 × (-1064)
Positive × Negative = Negative
348 × (-1064) = -370272
(f) 348 × 964
Positive × Positive = Positive
348 × 964 = 335472
Arranging in increasing order:
(e) 370272, (a) -1412, (b) 716, (c) 1412, (f) 335472, (d) 370272
Hence, (e)

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 13

Given that (-548) × 972 = -532656, write the values of:
(a) (-547) × 972
(b) (-548) × 971
(c) (-547) × 971

Accepted Answer

(a) Given: (-547) × 972 = -532656
We need to find (-547) × 972 = (-548 + 1) × 972
= -548 × 972 + 972
= -532656 + 972
= -531684
(b) Now -548 × 971 = -548 × (972 – 1)
= -548 × 972 – 548 × (-1)
= -532656 + 548
= -532108
(c) Here -547 × 971 = (-548 + 1) × (972 – 1)
= -548 × 972 + 548 + 972 – 1
= -532656 + 548 + 972 – 1
= -531137

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 14

Given that 207 × (-33 + 7) = -5382, write the value of -207 × (33 – 7) = ________

Accepted Answer

Here 207 × (-33 + 7)
= 207 × (-26)
= -5382
Now -207 × (33 – 7)
= -207 × 26
= -(207 × 26)
= -5382

Source: This question is from Operations with Integers, Mathematics &mdash; Class 7, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Operations with Integers: Value, Operations with Integers, Mathematics. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 7 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Operations with IntegersQ1: Let us try to find a few more pairs of numbers from their sums and differences: (a) Sum = 27,...
Q10: An alien society uses a peculiar currency called ‘pibs’ with just two denominations of coins — a +...
Q11: Find the values of: (a) (32 × (-18)) ÷ ((-36)) (b) (32) ÷ ((-36) × (-18)) (c) (25 × (-12)) ÷ ((45)...
Q12: Arrange the expressions given below in increasing order: (a) (-348) + (-1064) (b) (-348) – (-1064)...
Q13: Given that (-548) × 972 = -532656, write the values of: (a) (-547) × 972 (b) (-548) × 971 (c)...
Practice more questions from Operations with Integers &mdash; Mathematics, Class 7 CBSE

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 15

Use the numbers 3, -2, 5, -6 exactly once and the operations ‘+’, ‘-’, and ‘×’ exactly once and brackets as necessary to write an expression such that-
(a) The result is the maximum possible
(b) The result is the minimum possible

Accepted Answer

(a) Maximum possible value = [(-2) – (3 + 5)] × (-6)
= (-2 – 8) × (-6)
= 60
(b) Minimum possible value = [3 – (-2) + 5)] × (-6)
= (3 + 2 + 5) × -6
= -60

Source: This question is from Operations with Integers, Mathematics &mdash; Class 7, CBSE Board.
Key Concepts CoveredThis question tests your understanding of the following concepts from the chapter Operations with Integers: Use, Numbers, Exactly, Once, Operations, Brackets. These are fundamental topics in Mathematics that students are expected to master as part of the CBSE Class 7 curriculum.A thorough understanding of these concepts will help you answer similar questions confidently in your CBSE examinations. These topics are frequently tested in both objective and subjective sections of Mathematics papers. We recommend revising the relevant section of your textbook alongside practising these solved examples to build a strong foundation.
How to Approach This QuestionRead the question carefully and identify what is being asked. Break down complex questions into smaller parts. Use the terminology and concepts discussed in this chapter. Structure your answer logically — begin with a definition or key statement, then provide supporting details. Review your answer to ensure it addresses all parts of the question completely.
Key Points to RememberAlways show your working steps clearly.
Verify your answer by substituting values back into the equation.
Practice similar problems from the textbook exercises.
Memorise important formulae and their conditions of applicability.
Related Questions from Operations with IntegersQ1: Let us try to find a few more pairs of numbers from their sums and differences: (a) Sum = 27,...
Q10: An alien society uses a peculiar currency called ‘pibs’ with just two denominations of coins — a +...
Q11: Find the values of: (a) (32 × (-18)) ÷ ((-36)) (b) (32) ÷ ((-36) × (-18)) (c) (25 × (-12)) ÷ ((45)...
Q12: Arrange the expressions given below in increasing order: (a) (-348) + (-1064) (b) (-348) – (-1064)...
Q13: Given that (-548) × 972 = -532656, write the values of: (a) (-547) × 972 (b) (-548) × 971 (c)...
Practice more questions from Operations with Integers &mdash; Mathematics, Class 7 CBSE

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.

Question 16

Fill in the blanks in at least 5 different ways with integers:

Accepted Answer

(a) Here
(i) 0 + -6 × 6 = -36
(ii) 4 + (-5 × 8) = -36
(iii) -4 + -8 × 4 = -36
(iv) 12 + -8 × 6 = -36
(v) -3 + 3 × (-11) = -36
(b) Here
(i) (13 – 1) × 1 = 12
(ii) (10 – 4) × 2 = 12
(iii) (1 – 3) × (-6) = 12
(iv) (14 – 10) × 3 = 12
(v) (16 – 13) × 4 = 12
(c) Here
(i) (5 – (10 – 4)) = -1
(ii) (0 – (3 – 2)) = -1
(iii) (-1 – (1 – 1)) = -1
(iv) (-5 – (0 – 4)) = -1
(v) (-10 – (-5 – 4)) = -1

Chapter Overview: Integers
    This chapter covers multiplication and division of integers, extending the operations learned in Class VI. Students learn sign rules for multiplication and division, properties like closure, commutative, associative, and distributive, and apply integer operations to real-world contexts like temperature changes and financial transactions.
    Exam Weightage: ~6 marks | Difficulty: Medium

Key Formulas

Formula
        When to Use

(+) × (+) = (+), (+) × (-) = (-)Sign rule for multiplication
      (-) × (+) = (-), (-) × (-) = (+)Sign rule for multiplication
      a × (b + c) = a×b + a×cDistributive property
      a × 1 = a; a × 0 = 0Identity and zero property
      Division sign rules same as multiplicationDivision of integers

Must-Know Concepts
  
    Sign rules: same signs → positive, different signs → negative
    Closure property holds for multiplication but NOT for division:
    Commutative and associative properties hold for multiplication but NOT for division:
    Division by zero is undefined:
    Distributive property: a × (b + c) = a×b + a×c

Common Mistakes to Avoid
    
      Writing (-2) × (-3) = -6 instead of +6
      Claiming division is commutative
      Dividing by zero and writing 0 as the result

Scoring Tips
    
      Master sign rules first — they apply to both multiplication and division
      Use the pattern approach to understand why negative × negative = positive
      Practice with number line for visual understanding
      Memorize: even number of negatives = positive result

Frequently Asked Questions
  
    Why is negative × negative = positive?
    Look at the pattern: (-3)×3=-9, (-3)×2=-6, (-3)×1=-3, (-3)×0=0. Each step adds 3. So (-3)×(-1)=3, (-3)×(-2)=6. The pattern forces the result to be positive.

Why can't we divide by zero?
    Division is the inverse of multiplication. If 6÷0 = x, then x×0 should equal 6. But anything times 0 is 0, never 6. So no answer exists — division by zero is undefined.

Does closure hold for division of integers?
    No. For example, 7÷2 = 3.5, which is not an integer. So integers are not closed under division.