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.
$\blacksquare$
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 $2 \text{-} 1$