Greatest Common Divisor of Integers/Examples/8 and 17

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.

$\blacksquare$

Sources

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Example $2 \text{-} 1$