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Question:

Let $n\in\mathbb{N}$ and $p$ prime

i) Show $\forall h\in\mathbb{N}$, the number of $m\in\mathbb{Z}$, with $1\leq m\leq n$, divisible by $p^h$ is

equal to $\left[\frac{n}{p^h}\right]$, where $\left[\theta\right]$ denotes the integer part of $\theta$

ii) Hence deduce that the power of $p$ dividing $n!$ is $p^{e_p}$, where $e_p<\frac{n}{p-1}$

I am fine with part (i) so we can assume that answer.

Part (ii)

Answer (ii):

(1) For each $h\in\mathbb{N}$, there are $\left[\frac{n}{p^h}\right]−\left[\frac{n}{p^{h+1}}\right]$ such integers divisible by $p^h$ but not by $p^{h+1}$.

(2) Also, when h is sufficiently large, one has $\left[\frac{n}{p^h}\right]=0$.

(3) Then the largest power of $p$ dividing $n!$, ($e_p$), is: $$\sum_{m=0}^\infty m\left(\left[\frac{n}{p^m}\right]-\left[\frac{n}{p^{m+1}}\right]\right)$$

(4) $$\sum_{m=0}^\infty m\left(\left[\frac{n}{p^m}\right]-\left[\frac{n}{p^{m+1}}\right]\right)=\sum_{m=1}^\infty\left[\frac{n}{p^m}\right]<\sum_{m=1}^\infty\frac{n}{p^m}=\frac{n}{p-1}$$

I understand most of this argument, but am comfused by part (3), why is there an $m$ before the

"$\left(\left[\frac{n}{p^m}\right]-\left[\frac{n}{p^{m+1}}\right]\right)$" part.

Also why do we take the sum of the number of integers divisible by $p^h$ but not by $p^{h+1}$?

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Those multiples of $p$ but not $p^2$ each give one power of $p$ to the product.
Those multiples of $p^2$ but not $p^3$ each give two powers of $p$.
Those multiples of $p^3$ but not $p^4$ each give three powers of $p$.
In (3), you have $\lfloor\frac{n}{p^m}\rfloor-\lfloor\frac{n}{p^{m+1}}\rfloor$ factors, each contribute $m$ powers of $p$ to the product.

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