# Prove Each of the Following:### a. \[(\gcd(a, b) = 1 \land \gcd(a, c) = 1) \Rightarrow \gcd(a, bc) = 1\]**Hint:** From the given $\exists s, t, u, v \in \mathbb{Z}: sa + tb = 1, \; ua + vc = 1$. Multiply and rearrange terms.### b. \[(a \mid bc \land \gcd(a, b) = 1) \Rightarrow a \mid c\]### c.\[\gcd(a, b) = \gcd(b, \text{rem}(a, b)), \; \text{where } \text{rem}(a, b) \text{ is the remainder of } a \text{ divided by } b.\]

a) To prove : For any three given integers, a, b, c if gcd(a, b) = 1 and gcd(a, c) = 1. then…

a. (gcd(a,b) = 1 ^ gcd(a, c) = 1) = gcd (a, bc) = 1 (Hint: From the given 3s, t, u, v E Z: sa + tb = 1, ua + vc = 1. Multiply and rearrange terms.) b. (a|bc ^gcd(a, b) = 1) = a|c c. gcd(a, b) = gcd(b,rem(a, b), where rem(a, b) is the remainder of a divided by b.

a. (gcd(a,b) = 1 ^ gcd(a, c) = 1) = gcd (a, bc) = 1 (Hint: From the given 3s, t, u, v E Z: sa + tb = 1, ua + vc = 1. Multiply and rearrange terms.) b. (a|bc ^gcd(a, b) = 1) = a|c c. gcd(a, b) = gcd(b,rem(a, b), where rem(a, b) is the remainder of a divided by b.

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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Question

# Prove Each of the Following:

### a.
\[
(\gcd(a, b) = 1 \land \gcd(a, c) = 1) \Rightarrow \gcd(a, bc) = 1
\]
**Hint:** From the given $\exists s, t, u, v \in \mathbb{Z}: sa + tb = 1, \; ua + vc = 1$. Multiply and rearrange terms.

### b.
\[
(a \mid bc \land \gcd(a, b) = 1) \Rightarrow a \mid c
\]

### c.
\[
\gcd(a, b) = \gcd(b, \text{rem}(a, b)), \; \text{where } \text{rem}(a, b) \text{ is the remainder of } a \text{ divided by } b.
\]

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