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# Greatest common divisor

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The **greatest common divisor** (**gcd**) or ** Highest common factor ** (** HCF** )of two integers is the greatest (largest) number that divides both of the integers evenly. Euclid came up with the idea of GCDs.

## Algorithm

The GCD of any two positive integers can be defined as a recursive function: [math] gcd(u, v) = \begin{cases} gcd(v, u\mbox{ mod }v), & \mbox{if }v \gt 0 \\ u, & \mbox{if }v = 0 \end{cases} [/math]

## Examples

The GCD of 20 and 12 is 4, since 4 times 5 equals 20 and 4 times 3 equals 12. You can notice 3 and 5 have no common factor, hence their GCD is 1.

The GCD of 81 and 21 is 3.