Problem Solving

For any positive integer n, the length of n is definned as the number of prime factors whose product is n. For example, the length of 75 is 3; since 75 = 3 * 5 * 5. How many two digit positive integers have length 6?

A) None
B) one
C) two
D) three
E) four

the answer is C) two, can someone please explain this

Take the smallest prime integers and play.

2*2*2*2*2*2 =64
Since if you times by 2 once more, it will be 128, which is THREE digits instead of the two digits asked for in the question.

so that's one.

let's try the next prime up from 2, which is obviously 3.

2*2*2*2*2*3=96

so that's two.

if you multiply by more primes, the products will be > 2 digits.

Well, the smallest number that can have 6 prime factors is 2^6, which is 64.

The next smallest number would be 2^5 * 3, which is 96.

So, those are the only two numbers under 100 with six prime factors.

Hope this helps!