17. Let a and b be natural numbers such that a=b Prove each of the propo- sitions in Parts (a) through (d). (The results of Exercise (1) and Theorem 3.10 may be helpful.) (a) If a is even, then 4 divides a. * (b) If 4 divides a, then 4 divides b (c) If 4 divides b, then 8 divides a (d) If a is even, then 8 divides a (e) Give an example of natural numbers a and b such that a is even and a? = b³, but b is not divisible by 8 %3D
17. Let a and b be natural numbers such that a=b Prove each of the propo- sitions in Parts (a) through (d). (The results of Exercise (1) and Theorem 3.10 may be helpful.) (a) If a is even, then 4 divides a. * (b) If 4 divides a, then 4 divides b (c) If 4 divides b, then 8 divides a (d) If a is even, then 8 divides a (e) Give an example of natural numbers a and b such that a is even and a? = b³, but b is not divisible by 8 %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
17 C please
Transcribed Image Text:Prove that there exists a y, with 1 is4 such that y, M.
Hint: One way is to let ymax be the largest of y1. y2, y3. y4.
17. Let a and b be natural numbers such that a? = b. Prove each of the propo-
sitions in Parts (a) through (d). (The results of Exercise (1) and Theorem 3.10
may be helpful.)
(a) If a is even, then 4 divides a.
(b) If 4 divides a, then 4 divides b
(c) If 4 divides b, then 8 diyides a.
(d) If a is even, then 8 divides a
(e) Give an example of natural numbers a and b such that a is even and
a? = b3, but b is not divisible by 8
* 18. Prove the following proposition:
Let a and b be integers with a 0. If a does not divide b, then the
equation ax + bx + (b + a) = 0 does not have a solution that is a
natural number.
Hint: It may be necessary to factor a sum of cubes. Recall that
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