Exercise 22, § 1

Let a and b be two integers $> 0$, let d be the gcd of a and b, and let n be an integer $\geq 1$; put $a = da_1$ and $b = db_1$; show that the denominator in the irreducible expression for the fraction $a(a+b) \ldots (a+(n-1)b)/n!$ admits only prime factors which divide b, and are $\leq n$ (observe that if p is a prime number not dividing $b_1$, then $b_1$ is invertible mod $p^r$ for all $r > 0$). Hence obtain a direct proof of the fact that the binomial coefficients $n!/(rn!(n-rn)! )$ are integers.