HW11 sol.pdf - MATH 2345 section 54\/55 Homework 11 Solutions Q1 Prove that the product of any two rational numbers is a rational number(Direct Proof

# HW11 sol.pdf - MATH 2345 section 54/55 Homework 11...

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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 ࠵? 5 is an integer (it is a product of two integers), we have that ࠵? 5 = ࠵? 7 . ࠵? 5 for some integer ࠵?′ . By definition of divisibility, we have