# Power of a number in the difference of two factorials

What is the highest power of 3 available in $58! - 38!$ ( ! stands for factorial)

I can take $38!$ out as common to get $38! ( \frac{58!}{38!} - 1).$ I know how to find out the power of 3 in $38!$ But it is the difference term inside the brackets which I am not able to handle. What power of 3 will be contained in that term?

Is my approach correct in the first place? If yes then how to proceed further and if not then what approach should I take.

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## 1 Answer

Hint: $$\frac{58!}{38!} = 3\cdot 13\cdot\frac{58!}{39!}.$$

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Let's see if I have got it right. The expression essentially becomes 38!*( 3n-1) where n is any integer. But 3n-1 cannot contain any power of 3 so the only power of 3 in the whole expression will be contained in 38!. 38! contains 17 powers of 3 so that is the power of 3 in the whole expression. – Suy Oct 5 '13 at 8:21
Absolutely. ${}$ – njguliyev Oct 5 '13 at 8:23