If R and S are different integers both divisible by 5, then which of the following is not necessarily true ?

(a) R - S is divisible by 5
(b) R + S is divisible by 10 ✓ UPSC's answer
(c) R × S is divisible by 25
(d) R² + S² is divisible by 5

Why the answer is (b)

• Let R = 5a and S = 5b, where a and b are distinct integers.

• For option (a), R - S = 5a - 5b = 5(a - b), which is always divisible by 5.

• For option (c), R × S = 25ab, which is always divisible by 25.

• For option (d), R² + S² = 25a² + 25b² = 25(a² + b²), which is divisible by 5 (and 25).

• For option (b), R + S = 5a + 5b = 5(a + b). This is divisible by 10 only if (a + b) is even. If a and b are both odd or both even, it works, but if one is odd and the other is even (e.g., R=5, S=10), the sum is 15, which is not divisible by 10.

• Therefore, R + S is not necessarily divisible by 10.

Why the other options are wrong

(a) R - S is divisible by 5: The difference of two multiples of 5 is always a multiple of 5, so this statement is necessarily true.
(c) R × S is divisible by 25: The product of two multiples of 5 contains a factor of 25, so this statement is necessarily true.
(d) R² + S² is divisible by 5: The sum of squares of two multiples of 5 is a multiple of 25, which is also a multiple of 5, so this statement is necessarily true.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.