Find Greatest Common Divisor of 520 and 314, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 520 $ by $ 314 $ and get the remainder
The remainder is positive ($ 206 > 0 $), so we will continue with division.
Step 2 :
Divide $ 314 $ by $ \color{blue}{ 206 } $ and get the remainder
The remainder is still positive ($ 108 > 0 $), so we will continue with division.
Step 3 :
Divide $ 206 $ by $ \color{blue}{ 108 } $ and get the remainder
The remainder is still positive ($ 98 > 0 $), so we will continue with division.
Step 4 :
Divide $ 108 $ by $ \color{blue}{ 98 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 5 :
Divide $ 98 $ by $ \color{blue}{ 10 } $ and get the remainder
The remainder is still positive ($ 8 > 0 $), so we will continue with division.
Step 6 :
Divide $ 10 $ by $ \color{blue}{ 8 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 7 :
Divide $ 8 $ 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.
| 520 | : | 314 | = | 1 | remainder ( 206 ) | ||||||||||||
| 314 | : | 206 | = | 1 | remainder ( 108 ) | ||||||||||||
| 206 | : | 108 | = | 1 | remainder ( 98 ) | ||||||||||||
| 108 | : | 98 | = | 1 | remainder ( 10 ) | ||||||||||||
| 98 | : | 10 | = | 9 | remainder ( 8 ) | ||||||||||||
| 10 | : | 8 | = | 1 | remainder ( 2 ) | ||||||||||||
| 8 | : | 2 | = | 4 | remainder ( 0 ) | ||||||||||||
| GCD = 2 | |||||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.