discrete math problem Prove the following statement using a

discrete math problem

Prove the following statement using a direct proof

If r and s are rational numbers, then the product of r and s is a rational number.

Solution

r and s are rational numbers so there exist integers, a,b and c,d, b and d non zero so that

(a,b)=1,(c,d)=1 and

r=a/b,s=c/d

rs=(ac)/(bd)

a and c are integers so ac is integer ,

Similarly bd is an integer ,hence rs is rational number