# Doubt in finding number of non-prime factors of an integer

The question is:

Find the number of non-prime factors of $4^{10} \times 7^3 \times 5^9$.

I represented the number as $2^{20} \times 7^3 \times 5^9$ then the number of factors of this integer is $21 \times 4 \times 10 = 840$ now I can only see that there could be only three prime factors here $2,7 \text{ and } 5$. So the number of non-prime factors should be $837$ but my module says the answer would be $437$, what exactly I am missing here?

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1 would be another factor to remove here that you are still counting in the 837 – JB King May 15 '14 at 6:57

One could speculate about why the person hired to solve the problems made the mistake. But note that $$11 \times 4 \times 10=440$$ so the person doing the solutions may not have noticed that $4$ is not prime. You did notice.
Your argument looks right to me. I think there is a misprint - either the $4$ in $4^{10}$ was supposed to be a prime number (probably $3$, given that $5$ was already in the factorization), or the $4$ in $437$ was supposed to be an $8$.