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