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Two-adic valuation measures divisibility, not ordinary magnitude

For a nonzero rational x, write x=2^v a/b with odd integers a,b; define v_2(x)=v, and set v_2(0)=infinity. Thus v_2(12)=2, v_2(3/8)=-3, and v_2(5/3)=0. Multiplication adds valuations; addition has valuation at least the smaller input valuation, with possible extra cancellation. For integers, v_2(x)>=p means x is zero modulo 2^p. Negative valuation indicates an uncanceled power of two in the denominator. An ordinary small number need not have high two-adic precision.

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Updated 2026-11-08

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