Prove that (x+1)(x+2)…(x+k) ≡0 (mod k) for all integers x given that k is a + integer.
x+1, x+2,..., x+k are k successive integers in a row. since the multiples of k occur every k integers, at least one (in fact, exactly one) of the x+somethings is a multiple of k.
thus the whole product is a multiple of k, which is to say, congruent to 0 mod k.