1. Prove each of the following theorems by using a direct proof. (a) The sum of three odd integers is an odd integer. (b) The quotient of two nonzero rational numbers is a rational number. 2. Use a proof by contrapositive to show that if m and n are integers and mn is even, then m is even or n is even.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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1. Prove each of the following theorems by using a direct proof.
(a) The sum of three odd integers is an odd integer.
(b) The quotient of two nonzero rational numbers is a rational number.
2. Use a proof by contrapositive to show that if m and n are integers and mn is even, then m is
even or n is even.
3. Prove each of the following theorems by using an appropriate method.
(a) If n is an even integer, then n³ is an even integer.
(b) If n³ + 5 is even, then n is odd.
(c) If m and n are both perfect squares, then mn is also a perfect square.
(d) If n = ab, where a and b are positive integers, then a ≤ √ñ or b ≤ √ñ.
Transcribed Image Text:1. Prove each of the following theorems by using a direct proof. (a) The sum of three odd integers is an odd integer. (b) The quotient of two nonzero rational numbers is a rational number. 2. Use a proof by contrapositive to show that if m and n are integers and mn is even, then m is even or n is even. 3. Prove each of the following theorems by using an appropriate method. (a) If n is an even integer, then n³ is an even integer. (b) If n³ + 5 is even, then n is odd. (c) If m and n are both perfect squares, then mn is also a perfect square. (d) If n = ab, where a and b are positive integers, then a ≤ √ñ or b ≤ √ñ.
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