Alternative solutions to two of the problems.

2.4.9. Suppose that a|c, and b|c, and (a,b)=1. Show that ab|c.

Since a|c, and b|c, we can write c=as, c=bt for some s, t in Z.

Since (a,b)=1 , we can write 1=am+bn for some m,n in Z.

Then c=cam+cbn=abmt+abns=ab(mt+ns). Hence ab|c as claimed.

2.4.19. Suppose that a|c, and b|c, and (a,b)=d. Show that ab|cd.

Since a|c, and b|c, we can write c=as, c=bt for some s, t in Z.

Since (a,b)=d , we can write d=am+bn for some m,n in Z.

Then cd=cam+cbn=abmt+abns=ab(mt+ns). Hence ab|cd as claimed.