(a) Two Multiples of 3 Examples: (i) (6, 9) LCM (6, 9) = 18 (ii) (9, 12) LCM (9, 12) = 36 (iii) (12, 18) LCM (12, 18) = 36 Observation: The LCM of two multiples of 3 is also a multiple of 3. Reason: Since both numbers are divisible by 3, their common multiples will also be divisible by 3. Hence, the LCM must include 3 as a factor. General Statement: The LCM of two multiples of 3 is always a multiple of 3. (b) Two Consecutive Even Numbers Examples: (i) (2, 4) LCM (2, 4) = 4 (ii) (6, 8) LCM (6, 8) = 24 (iii) (10, 12) LCM (10, 12) = 60 Observation: The LCM of two consecutive even numbers is half of their product. Reason: Consecutive even numbers always share a common factor of 2, but not more. Therefore, when finding the LCM, one factor of 2 overlaps, so the LCM becomes smaller than their product. General Statement: The LCM of two consecutive even numbers 2n and 2n + 2 is always equal to half of their product. or LCM (2n, 2n + 2) = \(\frac{2 n \times(2 n+2)}{2}\) = n(2n + 2) = 2n 2 + 2n (c) Two Consecutive Numbers Examples: (i) (7, 8) LCM (7, 8) = 56 (ii) (9, 10) LCM (9, 10) = 90 (iii) (10, 11) LCM (10, 11) = 110 Observation: The LCM of two consecutive numbers is equal to their product. Reason: Consecutive numbers have no common factors other than 1, so their product is the smallest number divisible by both. General Statement: The LCM of two consecutive numbers is their product. (d) Two Co-prime Numbers Examples: (i) (4, 9) LCM (4, 9) = 36 (ii) (5, 8) LCM (5, 8) = 40 (iii) (7, 10) LCM (7, 10) = 70 Observation: The LCM of two co-prime numbers is equal to their product. Reason: Co-prime numbers do not share any common factors except 1, so the smallest number that contains both is simply their product. General Statement: The LCM of two co-prime numbers is equal to their product. Note: Co-prime numbers are any two natural numbers that have no common factor other than 1. Figure It Out (Pages 63-64)