The number of positive integers which divide ( 2 5 ) ! are - Options: A . &nbsp213.33.52 B . &nbsp28.32.52 C . &nbsp211.32.5 D . &nbsp28.33.53

The number of positive integers which divide (25)! are -

Question

The number of positive integers which divide

(2^{5})!

are -

Options:

A . 213.33.52

B . 28.32.52

C . 211.32.5

D . 28.33.53

Answer: Option A
:
A

32! = 2^{31} {.3}^{14} . 5^{7} {.7}^{4} {.11}^{2} {.13}^{2} .17 .19 .23 .29 .31

(on prime factorization)
So it has

(31 + 1) (14 + 1) (7 + 1) (4 + 1) (2 + 1) (2 + 1) (1 + 1) (1 + 1) (1 + 1) (1 + 1) (1 + 1) = 2^{13} . 3^{3} {.5}^{2}

factors.
Hence, choice (a) is the right answer.

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