   Chapter 2.4, Problem 12E

Chapter
Section
Textbook Problem

Prove that if d = ( a ,   b ) , a | c , and b | c , then a b | c d .

To determine

To prove: If d=(a,b),a|c and b|c then ab|cd.

Explanation

Given information:

d=(a,b),a|c and b|c

Formula used:

Greatest Common Divisors:

Let a and b be the integers, at least one of them is non-zero. Then, there exists a unique greatest common divisor d of a and b. Moreover, d can be written as d=am+bn for integers m and n, and d is the smallest positive integer that can be written in this form.

Proof:

Let (a,b)=d therefore, by using theorem on greatest common divisors, there exists integers m and n such that am+bn=d.

Multiplying the equation by c.

Therefore, cam+cbn=cd

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Subtract: 2301506

Elementary Technical Mathematics

Find the derivatives of the functions in Problems 1-14. 13.

Mathematical Applications for the Management, Life, and Social Sciences

Simplify: a. 64x639y2x6;x0 b. (a3b4)2(ba)3

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Evaluate the integral. 12x21xdx

Calculus (MindTap Course List)

Find the general indefinite integral. 2t(1+5t)dt

Single Variable Calculus: Early Transcendentals, Volume I

True or False: A straight (simple) substitution may be used to evaluate 2+lnxxdx.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 