## Task B-4.4.

Determine the largest natural number $n$ such that $\frac{500!}{7^{n}}$ is a natural number.
## Solution.

It is necessary to count how many times the number 7 appears in the product $500!=1 \cdot 2 \cdot 3 \cdot 4 \cdots 500$.

The number of multiples of 7 in this product is $k$, where $7 k$ is the largest natural number such that $7 k \leqslant 500$.

Thus, the number of multiples of 7 in the product is 71.

Among the multiples of 7, there are also multiples of $7^{2}=49$, and there are a total of 10 of them.

Since among the multiples of 49 is the number $7^{3}=343$, we conclude that the factor $7$ appears $71+10+1=82$ times in the product.

The largest natural number $n$ such that $\frac{500!}{7^{n}}$ is a natural number is 82.