How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers ?
- (a) 6
- (b) 7
- (c) 8
- (d) 9 ✓ UPSC's answer
Why the answer is (d)
• If 1186 leaves remainder 31 on division by n, then n divides 1186 − 31 = 1155, and n must be greater than 31.
• 1155 = 3 × 5 × 7 × 11, which has 2 × 2 × 2 × 2 = 16 divisors.
• The divisors up to 31 are 1, 3, 5, 7, 11, 15 and 21 — seven of them.
• That leaves 16 − 7 = 9 divisors greater than 31: 33, 35, 55, 77, 105, 165, 231, 385 and 1155 — option (d).
Why the other options are wrong
- (a) 6
- Option (a) is incorrect because it undercounts the valid divisors of 1155 that are greater than 31.
- (b) 7
- Option (b) is incorrect because it fails to account for all 9 divisors of 1155 exceeding the remainder 31.
- (c) 8
- Option (c) is incorrect because it misses one of the valid divisors of 1155 that is greater than 31.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.