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Let a, b e Z and let d:= (a, b). Prove that { ia + jb : i,j € Z}={kd : k E Z}. In other words, prove that the integers of the form ia + jb are precisely the multiples of (a, b). (This result will be very important when we study the ideals of Z.)

Let a, b e Z and let d:= (a, b). Prove that { ia + jb : i,j € Z}={kd : k E Z}. In other words, prove that the integers of the form ia + jb are precisely the multiples of (a, b). (This result will be very important when we study the ideals of Z.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let a, b e Z and let d := (a, b). Prove that { ia + jb : i, j e Z}= {kd : k E Z}. In other words, prove that the integers of the form ia + jb are precisely the multiples of (a, b). (This result will be very important when we study the ideals of Z.) 2.
Transcribed Image Text:Let a, b e Z and let d := (a, b). Prove that { ia + jb : i, j e Z}= {kd : k E Z}. In other words, prove that the integers of the form ia + jb are precisely the multiples of (a, b). (This result will be very important when we study the ideals of Z.) 2.
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