Show that for any numbers a b c elementof Z if a b or a c

Show that for any numbers a, b, c elementof Z^+, if a | b or a | c, then a | bc. (similar to zyBooks exercise 4.7.2 (a)).

Solution

lets take a|b, so there exist a integer r such that b=ar.

then bc = arc = a(rc). we wriiten bc interms of a and some integer..

Since rc is an integer..., a|bc

Hence proved..

Lets take a|c, so there exist a integer r such that c=ar.

then bc = bar = a(br). we wriiten bc interms of a and some integer..

Since br is an integer.. a|bc

Hence proved