MATH 2345 section 54/55 Homework 11 Solutions Q1. Prove that the product of any two rational numbers is a rational number. (Direct) $$∎Q2. Prove that for all integers ࠵?and ࠵?, if ࠵?|࠵?then ࠵?5|࠵?5. (Direct) Proof. Let ࠵?and ࠵?be [particular but arbitrarily chosen] integers and ࠵?|࠵?. We must show that ࠵?5|࠵?5. Since ࠵?|࠵?, by definition of divisibility, ࠵? = ࠵?. ࠵?for some integer ࠵?. Raise both sides to power of 2. We have ࠵?5= ࠵?5. ࠵?5. Since ࠵?5is an integer (it is a product of two integers), we have that ࠵?5= ࠵?7. ࠵?5for some integer ࠵?′. By definition of divisibility, we have