Exercise 233Number theory › Divisibility and gcdHigh schoolAdded 3 Nov 2026

GCD of a GCD

#gcd#associativity#three numbers

Statement

Show that $\gcd(\gcd(a,b),c)=\gcd(a,\gcd(b,c))$ for all $a,b,c\in\mathbb{Z}$ .

Hint

Both sides equal $\gcd(a,b,c)$ , the greatest common divisor of the three integers.

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Solution

Definition. $\gcd(a,b,c)$ is the greatest positive integer dividing all three. The identity $\gcd(\gcd(a,b),c)=\gcd(a,b,c)$ is a standard property (each side divides the other).

Conclusion. The GCD operation is associative.

Application. $\gcd(a,b,c)=\gcd(\gcd(a,b),c)$ , allowing iterative computation.

GCD of a GCD

Statement

Solution

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