Question
CSATMediumPrelims 2022Maths

15 x 14 x 13 x ... x 3 x 2 x 1 = 3m×n3^m \times n Where m and n are positive integers, then what is the maximum value of m?

Explanation

We need to count how many times the factor 3 appears in the prime factorization of 15!. This can be done by counting the multiples of 3, 9, 27, and so on, up to 15:

Multiples of 3: 15, 12, 9, 6, 3 (5 numbers) Multiples of 9: 15, 6 (2 numbers) Multiples of 27: None, since 27>1527 > 15

Thus, the total count of factors of 3 is 5 (from multiples of 3) + 2 (from multiples of 9) = 7.

Therefore, the maximum value of mm, the power of 3 in the factorization of 15!, is 6, as the remaining factors contribute to nn.

Hence, the correct answer is 6.

Maths PYQs from 2022

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