Least Positive Composite Integer

GRE Quant Problem Solving
1

Question: What is the least positive composite integer (not a prime number) that is not a factor of 40!?

The given solution lists 82, as the first integer that is not a factor is 41. Then the solution simply multiplies by 41 by 2 to get the correct answer. However, I don’t understand why all other composite integers between 41 and 82 are easily excluded?

2

A number such as 42 (for example) is composed of smaller prime factors (2, 3, and 7), which are revealed through prime factorization. Since each of these individual primes is less than or equal to 40, they are already present as factors within 40! itself.

Ultimately, your goal is to identify a number that has at least one prime factor greater than 40.