Question

Find Greatest Common Divisor of 12345 and 54321, using Euclidean algorithm.

Answer

The GCD of given numbers is 3.

Explanation

Step 1 :
Divide $54321$ by $12345$ and get the remainder

54321 : 12345 = 4 ( remainder is 4941)

The remainder is positive ( $4941 > 0$ ), so we will continue with division.

Step 2 :
Divide $12345$ by $4941$ and get the remainder

12345 : 4941 = 2 ( remainder is 2463)

The remainder is still positive ( $2463 > 0$ ), so we will continue with division.

Step 3 :
Divide $4941$ by $2463$ and get the remainder

4941 : 2463 = 2 ( remainder is 15)

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

Step 4 :
Divide $2463$ by $15$ and get the remainder

2463 : 15 = 164 ( remainder is 3)

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

Step 5 :
Divide $15$ by $3$ and get the remainder

15 : 3 = 5 ( 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

12345

54321

823

953

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

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