TITLE: Determine whether $\frac{1000!}{100!^{10}}$ is an integer
QUESTION [9 upvotes]: Can you give an idea, how to find out whether the result of ${1000!}/{100!^{10}}$ an integer. 
Modulo division? But what I met was about powers like $2^{100}/125$...

REPLY [2 votes]: I have a stronger statement:   
$$\binom{1000}{100}=\frac{1000!}{100!^{10}\binom{900}{100}\binom{800}{100}\binom{700}{100}\binom{600}{100}\binom{500}{100}\binom{400}{100}\binom{300}{100}\binom{200}{100}}$$   
is an integer. More generally:    
$$\binom{ab}{b}=\frac{(ab)!}{b!^{a}\binom{(a-1)b}{b}\binom{(a-2)b}{b}\cdots\binom{2b}{b}}$$   
is an integer (for $a\ge 2, b\ge 0$).