421 and 427, when divided by the same number, leave the same remainder 1. How many numbers can be used as the divisor in order to get the same remainder 1?
- (a) 1
- (b) 2
- (c) 3 ✓ UPSC's answer
- (d) 4
Why the answer is (c)
• If a number d leaves remainder 1 when dividing both 421 and 427, then d divides 420 and 426, and d must be greater than 1.
• The common divisors of 420 and 426 are the divisors of HCF(420, 426) = 6: 1, 2, 3, 6.
• 1 is excluded (it leaves remainder 0). 2, 3 and 6 each leave remainder 1 with both 421 and 427.
• So 3 divisors are possible — option (c).
Why the other options are wrong
- (a) 1
- There are three valid divisors (2, 3, and 6), not just one.
- (b) 2
- There are three valid divisors (2, 3, and 6), not just two.
- (d) 4
- The divisor 1 is invalid because it leaves a remainder of 0, not 1, so there are only three valid divisors.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.