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The binomial coefficients $\binom{n}{ p}$ are divisible by a prime $p$ only if $n$ is a power of $p.$

I'm looking for a "high school / undergraduate" demonstration for the: All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ ...
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highest power of prime $p$ dividing $\binom{m+n}{n}$

How to prove the theorem stated here. Theorem. (Kummer, 1854) The highest power of $p$ that divides the binomial coefficient $\binom{m+n}{n}$ is equal to the number of "carries" when adding $m$ ...
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Prove the following : [duplicate]

Prove the following : $${{n}\choose{7}}-\left \lfloor{\frac{n}{7}}\right \rfloor$$ is divisible by 7.
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Proof on Divisibility of Binomial Coefficients

Prove that $\exists \ i$ $(0 \lt i \lt n)$ such that $$n \nmid {n \choose i}$$ $\forall \ n$ such that $n \gt 0$ and $n$ is a composite Number.
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Simple method for $\frac{(2n+1)!}{(n!)^{2}}$ divide $lcm(1,2,\ldots,2n+1)$

The question is to prove that $\frac{(2n+1)!}{(n!)^{2}}$ divides $lcm(1,2,\ldots,2n+1)$. This seems like it should be a simple question, but try as I might, I can't seems to find any way that does ...
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How many numbers $k$ of $200 \choose k$ are divisible by $3$? $k \in \{0,1,2,\cdots 200\}$

"How many of the numbers (200 Choose k), where k is an element of the set {0,1,2,3,4,....,200} are divisible by 3? " Here is my thinking: (200 Choose 0,1, and 2) are not multiples of 3 but every ...
let $p$ is prime number,and such $p\mid n,p\nmid m,n\ge m$ show that $$p\>\Big|\>\binom{n}{m}$$ I know that: if $p$ is prime number,then $$\binom{n}{p}\equiv \left[\dfrac{n}{p}\right] \pmod ... 1answer 107 views Understanding a proof of the fact that \binom{n}{k} is always a natural number. Original source of question and solution. Question is on the left, answer is on the right. Question: Notice that all the numbers in Pascal's triangle are natural numbers. Use part (a) to prove by ... 3answers 160 views How can we find the gcd for elements (binomial coefficient)? \gcd\left(\binom{2n}1,\binom{2n}3,\binom{2n}5,\ldots,\binom{2n}{2n-1}\right) i want to know what is specialty of such a series.I am not able to generalize the problem solution.Is there any rule for ... 2answers 80 views Arithmetical proof of \cfrac{1}{a+b}\binom{a+b}{a} is an integer when (a,b)=1 When (a,b)=1, \cfrac{1}{a+b}\binom{a+b}{a} refers to the number of paths from one corner to its opposite corner of an a\times b lattice that lies completely above (or below) the diagonal. ... 3answers 118 views Combinatorial interpretation of this number? It is straightforward to show that if m,n\in\mathbb{Z} and m\geq n, then$$m\mid \gcd(m, n)\binom{m}{n}.$$I'm trying to find a combinatorial interpretation of this fact, but I can't seem to come ... 4answers 371 views Prove that \dfrac{(n^2)!}{(n!)^n} is an integer for every n \in \mathbb{N} Prove that \dfrac{(n^2)!}{(n!)^n} is an integer for every n \in \mathbb{N} I know that there are tools in Number theory to proves the required but I want to use the tool that says that if you ... 6answers 228 views How to simplify this expression by division How to divide it$${\frac {{x}^{n-2}-{y}^{n-2}}{x-y}}$$to remove the x-y term from the denominator. We may assume that n>2 is an integer. Thanks. 2answers 733 views Trying to prove that p prime divides \binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1 So I'm trying to prove that for any natural number 1\leq k<p, that p prime divides:$$\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1 Writing these choice functions in ...
Looks very easy, but I can't make it: $s \geq 2$ and $w \geq 2$ are prime numbers. $k$ is a natural number and $k \leq \min \{s,w \}$ Show that $\binom{s+w}{k}-\binom{w}{k} - \binom{s}{k}$ can be ...