Find Greatest Common Divisor of 34 and 56, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 56 $ by $ 34 $ and get the remainder
The remainder is positive ($ 22 > 0 $), so we will continue with division.
Step 2 :
Divide $ 34 $ by $ \color{blue}{ 22 } $ and get the remainder
The remainder is still positive ($ 12 > 0 $), so we will continue with division.
Step 3 :
Divide $ 22 $ by $ \color{blue}{ 12 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 4 :
Divide $ 12 $ by $ \color{blue}{ 10 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 5 :
Divide $ 10 $ by $ \color{blue}{ 2 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 2 }} $.
We can summarize an algorithm into a following table.
| 56 | : | 34 | = | 1 | remainder ( 22 ) | ||||||||
| 34 | : | 22 | = | 1 | remainder ( 12 ) | ||||||||
| 22 | : | 12 | = | 1 | remainder ( 10 ) | ||||||||
| 12 | : | 10 | = | 1 | remainder ( 2 ) | ||||||||
| 10 | : | 2 | = | 5 | remainder ( 0 ) | ||||||||
| GCD = 2 | |||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.