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(a) Prove that any multiple of d is an integer linear combination of a and b. (b) Prove that d divides any integer linear combination of a and b.

(a) Prove that any multiple of d is an integer linear combination of a and b. (b) Prove that d divides any integer linear combination of a and b.

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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Problem 2: The goal of this problem is to find all the numbers that can be written as integer linear combinations of two positive integers a and b. Let d be the smallest positive number that is an integer linear combination of a and b. Recall that d is also the greatest common divisor of a and b (see HW 2, Problem 6). (a) Prove that any multiple of d is an integer linear combination of a and b. (b) Prove that d divides any integer linear combination of a and b. Remark: Together these two results mean that the integer linear combinations of a and b are exactly the multiples of d.
Transcribed Image Text:Problem 2: The goal of this problem is to find all the numbers that can be written as integer linear combinations of two positive integers a and b. Let d be the smallest positive number that is an integer linear combination of a and b. Recall that d is also the greatest common divisor of a and b (see HW 2, Problem 6). (a) Prove that any multiple of d is an integer linear combination of a and b. (b) Prove that d divides any integer linear combination of a and b. Remark: Together these two results mean that the integer linear combinations of a and b are exactly the multiples of d.
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