0
votes
1answer
56 views

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 $p$ ...
2
votes
2answers
30 views

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 $m$ ...
2
votes
0answers
55 views

Prove the following : [duplicate]

Prove the following : $$ {{n}\choose{7}}-\left \lfloor{\frac{n}{7}}\right \rfloor $$ is divisible by 7.
1
vote
1answer
35 views

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.
4
votes
1answer
84 views

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 ...
4
votes
3answers
120 views

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 ...
3
votes
3answers
122 views

How prove this $\binom{n}{m}\equiv 0\pmod p$

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 ...
-1
votes
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 ...
8
votes
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 ...
3
votes
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. ...
4
votes
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 ...
17
votes
4answers
369 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 ...
1
vote
6answers
225 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.
4
votes
2answers
725 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 ...
4
votes
2answers
248 views

Binomial division

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 ...