यदि R और S दोनों अलग-अलग पूर्ण संख्याएँ हों और दोनों 5 से विभाज्य हों तो इनमें से कौन-सा अनिवार्यतः सही नहीं है ?
Read this question in English
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 · (c) R × S is divisible by 25 · (d) R² + S² is divisible by 5
उत्तर और व्याख्या · explanation in English
Answer: (b) R + S, 10 से विभाज्य है
Write R = 5a and S = 5b. The difference, product and sum of squares always keep the factor 5 (or 25), but R + S = 5(a + b) is a multiple of 10 only when a + b is even — 5 + 10 = 15 shows it can fail.
- Let R = 5a and S = 5b, where a and b are different integers.
- R − S = 5(a − b), R × S = 25ab and R² + S² = 25(a² + b²): always true.
- R + S = 5(a + b) is divisible by 10 only if a + b is even. Counter-example: R = 5, S = 10 gives 15.
- ✗ (a) R − S = 5(a − b) is always a multiple of 5.
- ✓ (b) Fails for R = 5, S = 10: the sum 15 is not divisible by 10.
- ✗ (c) R × S = 25ab is always a multiple of 25.
- ✗ (d) R² + S² = 25(a² + b²) is always a multiple of 5.
Remember · For 'not necessarily true', try one quick counter-example with the smallest numbers allowed.
Question and answer: UPSC's official GS Paper II (2016, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·