Question

证明若a,b和c是整数且c|ab,则c|(a,c)(b,c)

Answer 1

用到最大公约的性质:(x,y)z = (xz,yz).

于是(a,c)(b,c) = (a(b,c),c(b,c)) = ((ab,ac),c(b,c)).

由c | ab,可知c | (ab,ac),又显然c | c(b,c),故c | ((ab,ac),c(b,c)) = (a,c)(b,c).