Find Greatest Common Divisor of 312 and 418, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 418 $ by $ 312 $ and get the remainder
The remainder is positive ($ 106 > 0 $), so we will continue with division.
Step 2 :
Divide $ 312 $ by $ \color{blue}{ 106 } $ and get the remainder
The remainder is still positive ($ 100 > 0 $), so we will continue with division.
Step 3 :
Divide $ 106 $ by $ \color{blue}{ 100 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 4 :
Divide $ 100 $ by $ \color{blue}{ 6 } $ and get the remainder
The remainder is still positive ($ 4 > 0 $), so we will continue with division.
Step 5 :
Divide $ 6 $ by $ \color{blue}{ 4 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 6 :
Divide $ 4 $ 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.
| 418 | : | 312 | = | 1 | remainder ( 106 ) | ||||||||||
| 312 | : | 106 | = | 2 | remainder ( 100 ) | ||||||||||
| 106 | : | 100 | = | 1 | remainder ( 6 ) | ||||||||||
| 100 | : | 6 | = | 16 | remainder ( 4 ) | ||||||||||
| 6 | : | 4 | = | 1 | remainder ( 2 ) | ||||||||||
| 4 | : | 2 | = | 2 | remainder ( 0 ) | ||||||||||
| GCD = 2 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.