Question

Determinati numerele prime p,q şi r pentru care pqr = 5 x (p + 9 + r).
va rog frumos sa ma ajutati ! ​

Answer 1

$\displaystyle\bf\\pqr=5(p+q+r),~p,q,r~prime.\\pqr=5(p+q+r)=\mathcal{M}_5,~cum~p,q,r~sunt~prime,~singurul~numar~prim~este~5\\si~ecuatia~este~simetrica,~putem~considera~\boxed{\bf p=5}~.\\5qr=5(5+q+r) \Leftrightarrow qr=5+q+r\bigg|-(q+r) \implies qr-q-r=5,\\adunam~1~in~ambii~membri.\\qr-q-r+1=6,~putem~descompune~in~factori~membrul~stang.\\(r-1)(q-1)=6,~acum~il~scriem~pe~6~ca~produs~de~doua~numere~naturale,$

$\displaystyle\bf\\6=6\cdot1=1\cdot6=2\cdot3=3\cdot2,~ecuatia~fiind~simetrica,~este~suficient\\sa~analizam~cazurile~6=6\cdot1~si~6=2\cdot3.\\cazul~1,~6=6\cdot1 \implies \boxed{\bf r=7}~si~\boxed{\bf q=2}.\\cazul~2,~6=2\cdot3 \implies \boxed{\bf r=3}~si~\boxed{\bf q=4},~dar~q~nu~este~prim.\\deci,~avem~solutia~p=5,~q=2,~r=7,~si~\boxed{\bf permutarile}~!$