# Checking the divisibility of a factorial

Number Theory Level 5

When Madin learnt about factorials, he was delighted that they had so many factors since $$k! = 1 \times 2 \times 3 \times \ldots \times k$$. He knew that $$k!$$ would be divisible by any positive integer that was smaller than $$k$$, but wasn't certain about the larger integers.

Can you find the number of integers $$n$$ such that $$2\leq n\leq 100$$ and $$(n-1)!$$ is not divisible by $$n^2$$?

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