Greatest Common Divisor

The greatest common divisor of a set of integers is the largest number that divides each integer in the set. We denote the greatest common divisor by $\gcd(a, b, \ldots)$ . We can attempt to find this value by listing all divisors of the integers and finding the largest divisor. However, such a procedure can get tedious.

If the prime factorizations of $a$ and $b$ are

$\begin{aligned} a & = p_1 ^{\alpha_1} p_2 ^{\alpha_2} \ldots p_k ^{\alpha_k}, \\ b & = p_1 ^{\beta_1} p_2 ^ {\beta_2} \ldots p_k ^ {\beta_k}, \\ \end{aligned}$

then the GCD of the numbers is equal to

$\gcd(a,b) = p_1 ^{\min(\alpha_1, \beta_1)} p_2 ^{\min(\alpha_2, \beta_2)} \ldots p_k ^{\min(\alpha_k, \beta_k)} .$

For example: $\mbox{gcd}(12,18) = \mbox{gcd}(2^2 \cdot 3, 2 \cdot 3^2) = 2 \cdot 3 = 6$ .

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$