M31-11

Official Solution:

What is the least positive integer that is not a factor of 30! and is not a prime number?

A. 31
B. 32
C. 33
D. 62
E. 64


We are looking for a number that is not a factor of 30! and is also not a prime number.

The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is 2*31=62. Notice that all numbers between 30 and 62 are either primes (and we know that x is NOT a prime) or factors of 30! (because 30! has all its primes in higher powers). For example:

31 is a prime, hence \(x\) cannot be 31.

\(32 = 2^5\). Since 30! surely has 2 raised to a power greater than 5, 32 IS a factor of 30!.

\(33 = 3*11\). Both 3 and 11 are factors of 30!, hence 33 IS a factor of 30!

...

\(64 = 2^6\). Since 30! surely has 2 raised to a power greater than 6, 64 IS a factor of 30!.


Answer: D