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?
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.
If the question is "Find the number of non-prime factors of $~2^10 * 7^3 * 5^9~$"?

The answer will be $~440 - 3 =437~$