How to find the gcd of three numbers?

Understand the Problem

The question is asking how to calculate the greatest common divisor (gcd) for three given numbers, which involves identifying the largest positive integer that divides all three numbers without leaving a remainder.

Answer

The greatest common divisor is computed as $d = \text{gcd}(\text{gcd}(a, b), c)$.

Steps to Solve

Identify the three numbers We need to find the greatest common divisor (gcd) for the three given numbers. Let's denote them as $a$, $b$, and $c$.
Find the gcd of the first two numbers We can start by calculating the gcd of the first two numbers using the Euclidean algorithm. The gcd of two numbers $x$ and $y$ can be found as follows: $$ \text{gcd}(x, y) = \text{gcd}(y, x \mod y) $$ Continue this process until the remainder is zero.
Calculate gcd of the third number with the result Once we have the gcd of the first two numbers (let's call it $d$), we need to find the gcd of $d$ and the third number $c$. We can again use the Euclidean algorithm: $$ \text{gcd}(d, c) = \text{gcd}(c, d \mod c) $$ Continue this process until the remainder is zero.
Final result The result from the last gcd calculation will be the greatest common divisor of the three numbers $a$, $b$, and $c$.

Let $d = \text{gcd}(a, b)$ and then find $\text{gcd}(d, c)$. The final gcd will be the answer.

More Information

The greatest common divisor (gcd) is an important concept in number theory, often used in simplifying fractions and solving Diophantine equations. The Euclidean algorithm is a standard method for calculating gcd and is efficient for large integers.

Tips

Failing to apply the Euclidean algorithm correctly. Make sure to keep replacing the larger number with its remainder when divided by the smaller number.
Forgetting to find the gcd of the last number after calculating the first two. Remember to always complete the process with all three numbers.

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