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I took a competitive exam and there was this question

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

Can someone tell me how to find the answer.

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  • $\begingroup$ 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. $\endgroup$ – Don Thousand Apr 21 at 19:48
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Hint:

$$\sum_{k=1}^\infty\left\lfloor\frac{68}{7^k}\right\rfloor$$

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  • $\begingroup$ You should add the detail that it uses Legendre's formula for the $p$-valuation of $n!$, $p$ being a prime number. $\endgroup$ – Bernard Apr 21 at 19:55
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    $\begingroup$ I believe it should be the sum from $k=1$ $\endgroup$ – Will Jagy Apr 21 at 20:02
  • $\begingroup$ for example $7!$ should give exponent $1,$ while $6!$ should give exponent $0$ $\endgroup$ – Will Jagy Apr 21 at 20:06
  • $\begingroup$ @WillJagy Good catch, Will. Thanks $\endgroup$ – DonAntonio Apr 21 at 20:16

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