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# 4.4 Relatively Prime numbers

• Page ID
7591
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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.