Question

If p,r, and s are three different prime numbers greater than 2 and n = p*r*s, how many positive factors, including 1 and n, does n have?

Answer 1

8

here are two ways to look at it,

1. how many subsets can you form with 3 elements, including the null set (the factor 1), and the set of all elements (the n) -----> 2^3 = 8

or

2. we can use the p in 2 ways, that is, we can either include it or we can not include it, so 2 x ...
we can use the r in 2 ways, that is, we can either include it or we can not include it, so 2 x 2 x ..
we can use the s in 2 ways, that is, we can either include it or we can not include it, so 2 x 2 x 2

2 x 2 x 2 = 8,
this would include not taking any, which might correspond to the factor 1, and includes taking all 3 of them, which corresponds to n.

try it with the 3 numbers 3,5 and 7

factors would be
1
2
5
7
3*5
3*7
5*7 and finally
3*5*7, count them, there are 8