How many pairs of natural numbers are there such that the difference of whose squares is 63?
- (a) 3 ✓ UPSC's answer
- (b) 4
- (c) 5
- (d) 2
Why the answer is (a)
• x² − y² = (x − y)(x + y) = 63, with x > y natural numbers.
• Factor pairs of 63 with the smaller factor first: (1, 63), (3, 21), (7, 9).
• These give (x, y) = (32, 31), (12, 9) and (8, 1).
• So there are 3 pairs — option (a).
Why the other options are wrong
- (b) 4
- There are only 3 valid factor pairs of 63 that yield integer solutions for x and y, not 4.
- (c) 5
- The number of valid factor pairs is 3, not 5, as only three pairs of factors have the same parity.
- (d) 2
- There are 3 valid pairs, not 2, as the factor pair (7, 9) also yields a valid solution (8, 1).
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.