Let $p$ be a prime number. Prove that $p^{k}\choose n$ is divisible by $p$ for any $n$ where $1\le{n}\le{p^{k}}-1$ and $k$ is a positive integer.

Asked Nov 20 '20 at 19:00

Active Nov 20 '20 at 19:03

Viewed 105 times

Any and all help/hints/proofs would be appreciated, I really don't even know where to start.

edited Nov 20 '20 at 19:03

asked Nov 20 '20 at 19:00

itseasybutton

Fix title! It is not clear. – herb steinberg Nov 20 '20 at 19:03
@herbsteinberg done – itseasybutton Nov 20 '20 at 19:03
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This follows more or less immediately from Lucas Correspondence. For an on-site proof of that see e.g. here. Likely somebody else has written it more lucidly. – Jyrki Lahtonen Nov 20 '20 at 19:10
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Kummer's theorem on the $p$-adic value of binomial coefficients also implies your claim. – Jyrki Lahtonen Nov 20 '20 at 19:12
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Lest I forget. It looks like you may benefit from studying our guide for new askers. "I don't know where to start" -questions attract negative attention. – Jyrki Lahtonen Nov 20 '20 at 19:13
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The $28$ January $2019$ answer by bof to this question provides another answer to yours. – Brian M. Scott Nov 20 '20 at 19:34
@BrianM.Scott thank you so much – itseasybutton Nov 20 '20 at 19:42
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@itseasybutton: You’re very welcome. – Brian M. Scott Nov 20 '20 at 19:42
@JyrkiLahtonen thank you for your valuable information – itseasybutton Nov 20 '20 at 19:42

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