Ivan Shishkin, Rye (1878)

Problems/Number theoryExerciseUnreviewed

Some properties on the pp-adic valuation

by Sequoia·
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Let n,mn,m be two positive integers and pp be a prime number. Show the following properties of the pp-adic valuation:

  1. vp(nm)=m vp(n)v_p(n^m)=m\,v_p(n).
  2. vp(n+m)⩾min⁡(vp(n),vp(m))v_p(n+m)\geqslant \min(v_p(n),v_p(m)).
  3. vp(n+m)=min⁡(vp(n),vp(m))v_p(n+m)=\min(v_p(n),v_p(m)) if vp(n)≠vp(m)v_p(n)\neq v_p(m).
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