
# 4.4 Relatively Prime numbers

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Definition

Two integers are relatively prime when there are no common factors other than 1. This means that no other integer could divide both numbers evenly. Two integers $$a, b$$ are called relatively prime to each other if $$\gcd(a, b)=1$$.

For example, 7 and 20 are relatively prime.

Theorem

Let $$a, b\in \mathbb{Z}$$. If there exist integers $$x$$ and $$y$$ such that $$ax+by=1$$ then $$\gcd(a, b)=1$$.

Proof:

Let $$a, b\in \mathbb{Z}$$, such that d= $$\gcd(a, b)$$. Then d|a and d|b.

Hence d|(ax+by), thus d|1. Which implies d=+/- 1, since gcd is the greatest, d=1.

Example $$\PageIndex{1}$$:

Suppose $$a$$ and $$b$$ are relatively prime integers. Prove that $$\gcd(a+b,a-b)=1,$$ or $$2$$.

Example $$\PageIndex{2}$$:

Suppose $$a$$ and $$c$$ are relatively prime integers and $$b$$ is an integer such that $$b|c.$$ Prove that $$gcd(a,b)=1.$$