Question

Find Greatest Common Divisor of 123 and 456, using Euclidean algorithm.

Answer

The GCD of given numbers is 3.

Explanation

Step 1 :
Divide $456$ by $123$ and get the remainder

456 : 123 = 3 ( remainder is 87)

The remainder is positive ( $87 > 0$ ), so we will continue with division.

Step 2 :
Divide $123$ by $87$ and get the remainder

123 : 87 = 1 ( remainder is 36)

The remainder is still positive ( $36 > 0$ ), so we will continue with division.

Step 3 :
Divide $87$ by $36$ and get the remainder

87 : 36 = 2 ( remainder is 15)

The remainder is still positive ( $15 > 0$ ), so we will continue with division.

Step 4 :
Divide $36$ by $15$ and get the remainder

36 : 15 = 2 ( remainder is 6)

The remainder is still positive ( $6 > 0$ ), so we will continue with division.

Step 5 :
Divide $15$ by $6$ and get the remainder

15 : 6 = 2 ( remainder is 3)

The remainder is still positive ( $3 > 0$ ), so we will continue with division.

Step 6 :
Divide $6$ by $3$ and get the remainder

6 : 3 = 2 ( remainder is 0)

The remainder is zero => GCD is the last divisor $3$ .

We can summarize an algorithm into a following table.

This solution can be visualized using a Venn diagram.

0,0

123

456

The GCD equals the product of the numbers at the intersection.

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