The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?
- (a) 42
- (b) 38 ✓ UPSC's answer
- (c) 25
- (d) 16
Why the answer is (b)
• Let Mr. X's age last year be n² and his age next year be m³. Since these differ by 2 years, we have m³ − n² = 2, i.e., n² + 2 = m³.
• Testing small values of m: for m = 3, m³ = 27, so n² = 25 and n = 5. This gives last year's age = 25, current age = 26, and next year's age = 27 = 3³.
• The question asks how many years he must wait for his age to become a cube again, meaning the next perfect cube after 27. The next cube is 4³ = 64.
• From his current age of 26, the number of years until he reaches 64 is 64 − 26 = 38.
• Therefore, the least number of years he must wait is 38, which corresponds to option (b).
Why the other options are wrong
- (a) 42
- 26 + 42 = 68, which is not a perfect cube (4³ = 64 and 5³ = 125, so 68 falls between them).
- (c) 25
- 26 + 25 = 51, which is not a perfect cube (3³ = 27 and 4³ = 64, so 51 falls between them).
- (d) 16
- 26 + 16 = 42, which is not a perfect cube (3³ = 27 and 4³ = 64, so 42 falls between them).
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.