If 15 × 14 × 13 × ⋯ × 3 × 2 × 1 = 3^m × n where m and n are positive integers, then what is the maximum value of m?

(a) 7
(b) 6 ✓ UPSC's answer
(c) 5
(d) 4

Why the answer is (b)

• The expression 15 × 14 × 13 × ⋯ × 3 × 2 × 1 is the factorial of 15, denoted as 15!.

• To find the maximum value of m in 15! = 3^m × n, we need to determine the highest power of 3 that divides 15!.

• We use Legendre's Formula, which sums the integer divisions of 15 by powers of 3: floor(15/3) + floor(15/9) + floor(15/27) + ...

• Calculating the terms: floor(15/3) = 5, floor(15/9) = 1, and floor(15/27) = 0.

• Summing these values gives 5 + 1 = 6, so the maximum value of m is 6.

• Therefore, the correct option is (b).

Why the other options are wrong

(a) 7: Option (a) is incorrect because the sum of the exponents of 3 in the prime factorization of 15! is 6, not 7.
(c) 5: Option (c) is incorrect because it only accounts for the multiples of 3 (5) and misses the additional factor of 3 contributed by the multiple of 9.
(d) 4: Option (d) is incorrect because the calculated exponent of 3 in 15! is 6, which is greater than 4.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.