Soit \(a\), \(b\), \(c\) des entiers.
1
Montrer que \(\mbox{pgcd}(ca,cb)=\vert c\vert\;\mbox{pgcd}(a,b)\).
2
Montrer que \(\mbox{pgcd}(a^2,b^2)=(\mbox{pgcd}(a,b))^2\).
3
Montrer que si \(\mbox{pgcd}(a,b)=1\) et si \(c\) divise \(a\), alors \(\mbox{pgcd}(c,b)=1\).
4
Montrer que \(\mbox{pgcd}(a,bc)=1 \iff \mbox{pgcd}(a,b)=\mbox{pgcd}(a,c)=1\).
5
Montrer que si \(\mbox{pgcd}(b,c)=1\) alors \(\mbox{pgcd}(a,bc)=\mbox{pgcd}(a,b) \mbox{pgcd}(a,c)\).
6
Montrer que \(\mbox{pgcd}(a,b)=\mbox{pgcd}(a+b,\mbox{ppcm}(a,b))\).