By the unique prime factorization of integers every rational number x = 0 has a unique representation
x = ± ∏
pνp(x) , where νp(x) ∈ Z,
and the product is taken over all prime numbers p; but in fact only ﬁnitely many of the p-exponents νp(x) of x are non-zero. Thus, if we ﬁx a prime p, then there exist non-zero integers a, b for which
x = a
b · pνp(x) with ab ≡ 0 mod p.