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Conjecture: $\prod\limits_{k=0}^{n}\binom{2n}{k}$ is divisible by $\prod\limits_{k=0}^n\binom{2k}{k}$ only if $n=1,2,5$.
The diagram shows Pascal's triangle down to row $10$.
I noticed that the product of the blue numbers is divisible by the product of the orange numbers. That is (including the bottom centre number $...
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Divisibility property involving binomial coefficients and largest prime power divisor [duplicate]
Let $p$ be a prime, let $x$ be an integer not divisible by $p$, and let $j\geq 1$. Denote, as usual, by $\nu=\nu_p(j+1)$ the largest exponent such that $p^{\nu}$ divides $j+1$.
My question : is it ...
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For any prime number $p$ and natural number $i < p$, prove that $p$ divides ${p \choose i}$. [duplicate]
For any prime number $p$ and natural number $i < p$, prove that $p$ divides ${p \choose i}$.
Also, what happens when $p$ is not a prime. Is this still true?
I tried writing out the formula for ...
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