UPSC Prelims 2016·CSAT·Quantitative Aptitude·Number System

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

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  1. AR - S is divisible by 5
  2. BR + S is divisible by 10Correct
  3. CR × S is divisible by 25
  4. DR 2 + S 2 is divisible by 5

Explanation

Let R = 5k and S = 5m, where k and m are different integers. A) R - S is divisible by 5 R - S = 5k - 5m = 5(k - m). Since (k - m) is an integer, R - S is always divisible by 5. This is necessarily true. B) R + S is divisible by 10 R + S = 5k + 5m = 5(k + m). For this to be divisible by 10, (k + m) must be an even number. This means k and m must either both be even or both be odd. Consider a counterexample: Let R = 5 (so k = 1) and S = 10 (so m = 2). R and S are different integers, both divisible by 5. R + S = 5 + 10 = 15. 15 is not divisible by 10. Therefore, R + S is not necessarily divisible by 10. This is not necessarily true. C) R × S is divisible by 25 R × S = (5k) × (5m) = 25km. Since km is an integer, R × S is always divisible by 25. This is necessarily true. D) R^2 + S^2 is divisible by 5 R^2 + S^2 = (5k)^2 + (5m)^2 = 25k^2 + 25m^2 = 25(k^2 + m^2). Since 25(k^2 + m^2) is always divisible by 25, it is also always divisible by 5 (as 25 is a multiple of 5). This is necessarily true. The only statement that is not necessarily true is B.
Quantitative Aptitude: If R and S are different integers both divisible by 5, then which of the following is not necessarily true?
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