# Problem of the Day

How many distinct factors does have?

A. 2592

B. 2718

C. 3142

D. 3654

E. 3660

### Solution

[latexpage]

$14!=14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1$

Break down all of these numbers on the right into their prime factors:

$=(2*7)(13)(2^2*3)(11)(2*5)(3^2)(2^3)(7)(2*3)(5)(2^2)(3)(2)$

Simplify the expression using exponent laws:

$=2^{11}*3^5*5^2*7^2*11*13$

Any factor of this number must be of the form $2^a3^b5^c7^d11^e13^f$ where $a,b,c,d,e,f$ are integers and $0<=a<=11,0<=b<=5,0<=c<=2,0<=d<=2,0<=e<=1,0<=f<=1$.

Thus, when constructing a factor, we have 12 choices for $a$, 6 choices for $b$, 3 choices for $c$, 3 choices for $d$, 2 choices for $e$ and 2 choices for $f$.

Therefore there are $12*6*3*3*2*2=2592$ factors of $14!$

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