(i) Prime factorisation of 225 and 27 are 225 = 3 × 3 × 5 × 5
27 = 3 × 3 × 3
All prime factors of 27 are also prime factors of 225. But the prime factorisation of 12 is not included in the prime factorisation of 225.
This is because 3 occurs thrice in the prime factorisation of 27 but twice in the prime factorisation of 225. This means that 225 is not divisible by 27.
(ii) Prime factorisation of 96 and 24 are
96 = 2 × 2 × 2 × 2 × 2 × 3,
24 = 2 × 2 × 2 × 3
Since, we can multiply in any order,
96 = 24 × 4
Therefore, 96 is divisible by 24.
(iii) Prime factorisation of 343 and 17 are
343 = 7 × 7 × 7 and 17 = 17
The prime factorisation of 343 is 7 × 7 × 7 and 17 is a prime number with no common factor with 343.
Therefore, 343 is not divisible by 17.
(iv) Prime factorisation of 999 and 99 are
999 = 3 × 3 × 111,
99 = 3 × 3 × 11
Prime factorisation of 99 is not included in the prime factorisation of 999.
Therefore, 999 is not divisible by 99.