Problem
NT-B2-M05-P009 Product
#9
★★★☆☆ Level 3 of 5
Prove that \(v_p(ab)=v_p(a)+v_p(b)\) for prime \(p\).
Write \(a=p^r u\), \(b=p^s v\), where \(p\nmid u,v\).
Let \(a=p^r u\), \(b=p^s v\), where \(p\nmid u\) and \(p\nmid v\). Then \(ab=p^{r+s}uv\), and \(p\nmid uv\). Hence the exponent of \(p\) in \(ab\) is \(r+s\), namely \(v_p(a)+v_p(b)\).
Fundamental property of valuations.