Question

Prove that if a | c and b | c and gcd(a,b) = 1 then ab | c

Answer 1

if a|c then c=na
if b|c then c=mb
if (a,b)=1 then xa+yb=1

multiply xa+yb=1 by c

xac+ybc=c

replace c by mb and na

xamb+ybna=c

xmab+ynab=c

(xm+yn)(ab)=c

thus c is an integer multiple of ab. thus ab|c