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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
up vote 14 down vote accepted

You are missing making the error that your module made. You are right, the module is wrong.

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.

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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 $8$.

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