What is the largest integer \(n \) such that \(7^n\) divides \(1000!\)? Btw the answer isn't 163.
+501 We need to find how many factors of seven all the integers from 1 to 1000 have.
Each integer from 1 to 1000 can be a factor of 7, 49, or 343, giving it 1 factor of 7, 2 factors of 7, or 3 factors of 7.
There are \(\lfloor\frac{1000}{7}\rfloor=142\) numbers that have 1 factor of 7.
There are \(\lfloor\frac{1000}{49}\rfloor=20\) numbers that have 2 factors of 7.
There are \(\lfloor\frac{1000}{343}\rfloor=2\) numbers that have 3 factors of 7.
So, the answer is \(2\cdot3+20\cdot2+142\cdot1\).
But, wait!
We are counting numbers that have 2 factors in the numbers that have 1 factor of 7!
And, we are counting numbers that have 3 factors in the numbers that have 2 factors and the numbers that have 1 factor!
So, the answer is \(2\cdot3+20\cdot2+142\cdot1-2-2-20=2+20+142=164\)!
