how do i know if 7 is a factor of 5^2 * 31 * 43 * 10^6
You can calculate that expression and see if its divisble by 7.
So 5^2 * 31 * 43 * 10^6=33325000000 and divide that by 7 to see if integer comes up.
But another simpler way to see that 7 isn't factor is by looking at all factors at expression. So from expression we have factors 5,31,43 and 10 so 7 isn't factor.
As 7 is prime, if it is a factor of a product then it divides (is a factor of) at
least one of the terms of the product.
In this case if 7 is a factor of 5^2*31*43*10^6 it must be a factor of one
of:
5, 31, 43, 10,
which it is not so it is not a factor of 5^2*31*43*10^6.
Alternatively, break the product down to a product of prime factors,
if 7 appears among the list of prime factors then the 7 is a factor
of the product, otherwise it is not (the prime factorisation is unique
by the fundamental theorem of arithmetic).
RonL