- #1

Math100

- 793

- 220

- Homework Statement
- Find the highest power of ## 3 ## dividing ## 823! ##.

- Relevant Equations
- None.

Let ## n ## be a positive integer and ## p ## be a prime.

Then the exponent of the highest power of ## p ## that divides ## n! ## is ## \sum_{k=1}^{\infty}[\frac{n}{p^{k}}] ##.

Observe that ## n=823 ## and ## p=3 ##.

Thus

\begin{align*}

&[\frac{823}{3}]+[\frac{823}{3^{2}}]+[\frac{823}{3^{3}}]+[\frac{823}{3^{4}}]+[\frac{823}{3^{5}}]+[\frac{823}{3^{6}}]\\

&=274+91+30+10+3+1\\

&=409.\\

\end{align*}

Therefore, the highest power of ## 3 ## dividing ## 823! ## is ## 409 ##.

Then the exponent of the highest power of ## p ## that divides ## n! ## is ## \sum_{k=1}^{\infty}[\frac{n}{p^{k}}] ##.

Observe that ## n=823 ## and ## p=3 ##.

Thus

\begin{align*}

&[\frac{823}{3}]+[\frac{823}{3^{2}}]+[\frac{823}{3^{3}}]+[\frac{823}{3^{4}}]+[\frac{823}{3^{5}}]+[\frac{823}{3^{6}}]\\

&=274+91+30+10+3+1\\

&=409.\\

\end{align*}

Therefore, the highest power of ## 3 ## dividing ## 823! ## is ## 409 ##.