Question

Find Greatest Common Divisor of 267 and 165, using Euclidean algorithm.

Answer

The GCD of given numbers is 3.

Explanation

Step 1 :
Divide $267$ by $165$ and get the remainder

267 : 165 = 1 ( remainder is 102)

The remainder is positive ( $102 > 0$ ), so we will continue with division.

Step 2 :
Divide $165$ by $102$ and get the remainder

165 : 102 = 1 ( remainder is 63)

The remainder is still positive ( $63 > 0$ ), so we will continue with division.

Step 3 :
Divide $102$ by $63$ and get the remainder

102 : 63 = 1 ( remainder is 39)

The remainder is still positive ( $39 > 0$ ), so we will continue with division.

Step 4 :
Divide $63$ by $39$ and get the remainder

63 : 39 = 1 ( remainder is 24)

The remainder is still positive ( $24 > 0$ ), so we will continue with division.

Step 5 :
Divide $39$ by $24$ and get the remainder

39 : 24 = 1 ( remainder is 15)

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

Step 6 :
Divide $24$ by $15$ and get the remainder

24 : 15 = 1 ( remainder is 9)

The remainder is still positive ( $9 > 0$ ), so we will continue with division.

Step 7 :
Divide $15$ by $9$ and get the remainder

15 : 9 = 1 ( remainder is 6)

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

Step 8 :
Divide $9$ by $6$ and get the remainder

9 : 6 = 1 ( remainder is 3)

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

Step 9 :
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

267

165

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

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