In a certain code if 64 is written as 343 and 216 is written as 729, then how is 512 written in that code ?
- (a) 1000
- (b) 1331 ✓ UPSC's answer
- (c) 1728
- (d) 2197
Why the answer is (b)
• Identify the relationship between the given numbers: 64 is $4^3$ and 343 is $7^3$.
• Observe that the base of the cube increases by 3 (from 4 to 7), so the code is $(n+3)^3$ where $n$ is the cube root of the original number.
• Verify with the second pair: 216 is $6^3$ and 729 is $9^3$.
• The base increases by 3 (from 6 to 9), confirming the rule is to add 3 to the cube root and then cube the result.
• Apply the rule to 512: The cube root of 512 is 8.
• Add 3 to the base: $8 + 3 = 11$.
• Cube the new base: $11^3 = 1331$.
• Therefore, 512 is written as 1331, which corresponds to option (b).
Why the other options are wrong
- (a) 1000
- 1000 is $10^3$, which would imply adding 2 to the cube root (8+2), contradicting the established pattern of adding 3.
- (c) 1728
- 1728 is $12^3$, which would imply adding 4 to the cube root (8+4), contradicting the established pattern of adding 3.
- (d) 2197
- 2197 is $13^3$, which would imply adding 5 to the cube root (8+5), contradicting the established pattern of adding 3.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.