?

Abstract Algebra

Free Version

Upgrade subject to access all content

Moderate

Divisibility of Factorials and Binomial Coefficients by 499

ABSALG-3QYL4I

Which of the following numbers is divisible by the prime number $499$?

A calculator is not needed.

The notation $n!$ means $n$ factorial, defined as the product $1.2.\ldots.(n-1).n$ of all integers from $1$ to $n$.

The notation ${a\choose b}$, $1\le b\le a-1$, means the binomial coefficient ${a\choose b}=\cfrac{a!}{b!(a-b)!}$).

Select ALL that apply.

A

$499!$

B

$r!$ for any integer $1\le r\le 498$

C

$(499-r)!$ for any integer $1\le r\le 498$

D

${499\choose r}r!(499-r)!$ for any integer $1\le r\le 498$

E

${499\choose r}$, for any integer $1\le r\le 498$.