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Prove the following for positive integers, a, b, c and n: If an \equiv b(modc), then...

Question:

Prove the following for positive integers, a, b, c and n:

If {eq}an \equiv b(modc) {/eq}, then {eq}\frac{an}{gcd(a,c)} \equiv \frac{b}{gcd(a,c)}(mod\frac{c}{gcd(a,c)}) {/eq}

Greatest Common Divisor

This is a function defined over integers. It needs 2 integers and returns the greatest integer that can divide both the given numbers without a remainder. It is 1 for relatively prime numbers.

Answer and Explanation:

Given:

{eq}an \equiv b(modc) {/eq}

We can write an as such:

{eq}an = cx + b {/eq}

where x is integer.

Then, we can divide each side by an...

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Greatest Common Divisor: Definition & Formula

from General Studies Math: Help & Review

Chapter 1 / Lesson 6
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