The integers s and t are divisible by 5. Which of the following is NOT necessarily true?

- A s-t is divisible by 5.
- B s+t is divisible by 10.
- C st is divisible by 25.
- D is divisible by 5. s2−t2
- E is divisible by 25. s2+t2

The answer is B. But I thought I can be C also.

Can anyone please explain the answer?

We can consider s = 5m & t = 5n, where m & n are integers, since it is given to us that s & t are divisible by 5 (For eg 15 = 5*3 & 10 = 5*2).

For A: s - t = 5m - 5n = 5 (m-n) is divisible by 5, since (m-n) will also be an integer.

For B: s + t = 5m + 5n = 5 (m+n). Now, for (s + t) to be divisible by 10, we must have (m + n) to be a multiple of 2, ie either both odd or both even. As we don’t have any evidence about that, we cannot certainly say that 5 (m + n) would be divisible by 10.

For C: st = 5m*5n = 25mn. Since mn is an integer, st must be divisible by 25.

For D: s^2 - t^2 = (5m)^2 - (5n)^2 = 25 (m^2 - n^2). It will be divisible by 5 because we already have 25 (divisible by 5). Moreover, (m^2 - n^2) is an integer, which won’t affect the divisibility.

For E: s^2 + t^2 = (5m)^2 + (5n)^2 = 25 (m^2 + n^2). This will be divisible by 25. Moreover, (m^2 - n^2) is an integer, which won’t affect the divisibility.

According to me, B should be the correct choice.

Hope that helps.