If $p$ is a prime number greater than 2 and $k\in \mathbb{N}$ so that $k < p$, how can I prove that $p\choose k$ is congruent to $0 \bmod p$?
If $p$ is a prime number greater than $2$ and $k$ is a natural number so that $k
1
$\begingroup$
binomial-coefficients
modular-arithmetic
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1Note natural means $k \geq 1.$ This has probably been asked several times before, it's a standard and useful fact. – 2012-12-05
3 Answers
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Hint: Do you know that ${p \choose k}=\frac {p!}{k!(p-k)!}?$
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0I do I'm having trouble with the congruency part of it, I haven't done congruences in a while and was shaky at best when I did do them. – 2012-12-05
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0@JackForest: If $p$ is prime how can there be it a factor of the denominator? – 2012-12-05
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0There can not be a factor for the denominator since p is only divisible by 1 and itself. – 2012-12-05
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0@JackForest: spot on. Do you think $0 \in \mathbb N$? Then you need one more piece. – 2012-12-05
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00 is not ∈ N correct? – 2012-12-05
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0@JackForest: That depends on what part of math you are in. Some say yes, some no. So I believe you need to be explicit. But your remark indicates you understand the issue. – 2012-12-05
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0I wasn't sure if you were asking if I was of the belief that 0 was a natural number or not but now I understand your question, no we say that 0 is not a natural number. – 2012-12-05
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0How do I go about the congruence part of the question? – 2012-12-05
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0@JackForest: then both $k$ and $p-k$ are less than $p$ so there is a factor of $p$ in the numerator and not in the denominator. It is fun to look at Pascal's triangle mod $p$. – 2012-12-05
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So, you have: $$\frac{p!}{k!(p-k)!}$$ and you want to prove this is a multiple of $p$.
Hint: $n! = n(n-1)!$
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For $\,1\leq k\leq p-1\,$:
$$\binom{p}{k}=\frac{p!}{k!(p-k)!}=\frac{(k+1)(k+2)\cdot\ldots\cdot (p-1)}{1\cdot 2\cdot\ldots\cdot (p-k)}\cdot\;p$$
Since the binomial coefficient is an integer and all the factors both in the numerator and in the denominator above are smaller than $\,p\,$, we're done