# Remainder theorem of factorials and powers

If $$7^n$$ divides $$68!$$ Then what is the greatest value of $$n$$?

Can someone tell me how to find the answer.

• Basically, this is asking you, how many factors of 7 are there in 68!. DonAntonio's answer gives you a formula to compute this, but you should try to come up with an answer without the formula first to understand how it works. – Don Thousand Apr 21 at 19:48

$$\sum_{k=1}^\infty\left\lfloor\frac{68}{7^k}\right\rfloor$$
• You should add the detail that it uses Legendre's formula for the $p$-valuation of $n!$, $p$ being a prime number. – Bernard Apr 21 at 19:55
• I believe it should be the sum from $k=1$ – Will Jagy Apr 21 at 20:02
• for example $7!$ should give exponent $1,$ while $6!$ should give exponent $0$ – Will Jagy Apr 21 at 20:06