Focusing on v_x(p), the multiplicity of some prime p in the prime factorization of x: m = j p^a → v_m(p) = a n = k p^b → v_n(p) = b p is not a factor of

Focusing on v_x(p), the multiplicity of some prime p in the prime factorization of x: m = j p^a → v_m(p) = a n = k p^b → v_n(p) = b p is not a factor of j or k m^n = j^n (p^a)^n = j^n p^(an) = k^m p^(bm) = … = n^m v_{m^n}(p) = an = bm Note that a=0 iff b=0. This is the case where p is neither a factor of m nor n. b/a = m/n >=1 This ratio is the same for all primes, so every prime has: v_m(p) >= v_n(p) so n | m (We didn't need the ratio to be the same, but we did need the ratios to be all greater >= 1)