The exponent of 11 in the prime factorization of 300! is A. 27 B. 28 C. 29 D. 30

GATE 2008 · Discrete Mathematics · Number Theory · medium

Answer: The exponent of 11 in the prime factorization of 300! is 27 + 2 = 29 (option C).

1. Count multiples of 11 in {1, ..., 300}: floor(300 / 11) = floor(27.27...) = 27. These are 11, 22, 33, ..., 297; there are 27 of them.
2. Count multiples of 11^2 = 121 and higher powers: floor(300 / 121) = floor(2.479...) = 2 (the multiples 121 and 242 each contribute one extra factor of 11). floor(300 / 1331) = floor(0.225...) = 0. Higher powers also give 0. Total exponent = 27 + 2 + 0 = 29 (option C).