Greatest Common Divisor of Integers/Examples/20, 70 and 80

Example of Greatest Common Divisor of Integers

The greatest common divisor of $20$, $70$ and $80$ is:

$\gcd \set {20, 70, 80} = 10$

Proof

The strictly positive divisors of $20$ are:

$\set {x \in \Z_{>0}: x \divides 20} = \set {1, 2, 4, 5, 10, 20}$

The strictly positive divisors of $70$ are:

$\set {x \in \Z_{>0}: x \divides 70} = \set {1, 2, 5, 7, 10, 14, 35, 70}$

The strictly positive divisors of $80$ are:

$\set {x \in \Z_{>0}: x \divides 80} = \set {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}$

Of these, the common divisors are:

$\set {x \in \Z_{>0}: x \divides 20 \land x \divides 70 \land x \divides 80} = \set {1, 2, 5, 10}$

The greatest of these is $10$.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): common factor (common divisor)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): common factor (common divisor)