What is the largest integer $n$ such that $7^n$ divides $1000!$?

The correct answer is

n = 3038

Step-by-step explanation:

Where we have;

We note that 1000! = An even number

Since Even * Even = even

Even * Odd = Even

Also, 7ⁿ is an Odd number since odd * odd = odd

Therefore 7ⁿ will divide 1000! with some fractions as follows;

1000! divided by 7ⁿ

When 7ⁿ = 1000!

log (7ⁿ) = log (1000!) = log (4.02*10²⁵⁶⁷)

n·log (7) = 2567.604

n = 2567.604 / (log (7)) = 3038.23

Therefore, the largest integer n such that 7ⁿ divides 1000! = 3038

Which gives, 1000! : 7³⁰³⁸ = 1.5733.