4. What is the greatest power of 7 that divides the product $1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 999 \cdot 1000$?
4. S o l u t i o n. The number of integers divisible by 7 in the first thousand natural numbers is 142, those divisible by $49=7^{2}$ is 20, and those divisible by $343=7^{3}$ is 2. If the power of seven is greater than three, the corresponding power will be greater than 1000. Therefore, none of the factors of the product can be divisible by seven to a power greater than three. It is not difficult to show that the given product contains the number seven to the power of $142+20+2=164$.