Find Greatest Common Divisor of 81 and 51, using Euclidean algorithm.
Answer
The GCD of given numbers is 3.
Explanation
Step 1 :
Divide $ 81 $ by $ 51 $ and get the remainder
The remainder is positive ($ 30 > 0 $), so we will continue with division.
Step 2 :
Divide $ 51 $ by $ \color{blue}{ 30 } $ and get the remainder
The remainder is still positive ($ 21 > 0 $), so we will continue with division.
Step 3 :
Divide $ 30 $ by $ \color{blue}{ 21 } $ and get the remainder
The remainder is still positive ($ 9 > 0 $), so we will continue with division.
Step 4 :
Divide $ 21 $ by $ \color{blue}{ 9 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 5 :
Divide $ 9 $ by $ \color{blue}{ 3 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 3 }} $.
We can summarize an algorithm into a following table.
| 81 | : | 51 | = | 1 | remainder ( 30 ) | ||||||||
| 51 | : | 30 | = | 1 | remainder ( 21 ) | ||||||||
| 30 | : | 21 | = | 1 | remainder ( 9 ) | ||||||||
| 21 | : | 9 | = | 2 | remainder ( 3 ) | ||||||||
| 9 | : | 3 | = | 3 | remainder ( 0 ) | ||||||||
| GCD = 3 | |||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.