Greatest Common Divisor of Integers/Examples/8 and 17

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Example of Greatest Common Divisor of Integers

The greatest common divisor of $8$ and $17$ is:

$\gcd \set {8, 17} = 1$

That is, $8$ and $17$ are coprime.

Proof

The strictly positive divisors of $8$ are:

$\set {x \in \Z_{>0}: x \divides 8} = \set {1, 2, 4, 8}$

The strictly positive divisors of $17$ are:

$\set {x \in \Z_{>0}: x \divides 17} = \set {1, 17}$

It is seen that there is only one strictly positive common divisor of $8$ and $17$, and that is $1$.

Hence by definition $8$ and $17$ are coprime.

$\blacksquare$