UPSC CAPF (AC) 2025 Paper I · Question 57

Question 57 of the UPSC CAPF (AC) 2025 Paper I (3 August 2025). Try it first, then open the answer and explanation.

1. CAPF 2025 · P-I · Q57 · General Mental Ability · Numbers & Series

   If n is an integer larger than 1, then what is the least value of the integer n + n^(1/2) + n^(1/3)?

   1. 1
   2. 3
   3. 76
   4. 785

   यदि n, 1 से बड़ा पूर्णांक (integer) है, तो पूर्णांक n + n^(1/2) + n^(1/3) का न्यूनतम मान क्या है?

   1. 1
   2. 3
   3. 76
   4. 785
   Show answer

   Answer: (3) 76

   उत्तर: (3) 76

   For the sum to be an integer, n must be both a perfect square and a perfect cube, i.e. a perfect sixth power. The smallest sixth power greater than 1 is 2^6 = 64, giving 64 + 8 + 4 = 76 (1 and 3 are impossible because for any n ≥ 2 the sum already exceeds 4).

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