= Odd part of a positive integer
{title2=$n=2^{v_2(n)}d$}
= Odd part
{synonym}
= Odd part of an integer
{synonym}
The odd part of a positive <integer> $n$ is the unique odd positive <integer> $d$ obtained by repeatedly dividing $n$ by two. The process terminates because the positive <integers> decrease. If $2^r d=2^s e$ with $d,e$ odd, then $r>s$ would make $e$ even and $s>r$ would make $d$ even; hence $r=s$ and $d=e$. Two positive <integers> have the same odd part exactly when their ratio is an integral power of two. The odd positive <integers> therefore give representatives for the resulting <equivalence classes>.
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