Compute the number of perfect square factors in the base ten number , where n! represents n factorial.
.
Factoring, this is , or . A perfect square factor is going to have an even exponent in each of the prime factors. There are 5 possibilities for the exponent of 2 (0,2,4,6,8), 2 possibilities for 3, 2 possibilities for 5, and 4 possibilities for 7.
Hence the total number of square factors is .
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Factoring, this is![[7^2\cdot 5 \cdot 3 \cdot 2^3]^3](http://s0.wp.com/latex.php?latex=%5B7%5E2%5Ccdot+5+%5Ccdot+3+%5Ccdot+2%5E3%5D%5E3&bg=ffffff&fg=333333&s=0 "[7^2\cdot 5 \cdot 3 \cdot 2^3]^3")
, or 
. A perfect square factor is going to have an even exponent in each of the prime factors. There are 5 possibilities for the exponent of 2 (0,2,4,6,8), 2 possibilities for 3, 2 possibilities for 5, and 4 possibilities for 7.
Hence the total number of square factors is
.