Find Greatest Common Divisor of 457 and 378, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 457 $ by $ 378 $ and get the remainder
The remainder is positive ($ 79 > 0 $), so we will continue with division.
Step 2 :
Divide $ 378 $ by $ \color{blue}{ 79 } $ and get the remainder
The remainder is still positive ($ 62 > 0 $), so we will continue with division.
Step 3 :
Divide $ 79 $ by $ \color{blue}{ 62 } $ and get the remainder
The remainder is still positive ($ 17 > 0 $), so we will continue with division.
Step 4 :
Divide $ 62 $ by $ \color{blue}{ 17 } $ and get the remainder
The remainder is still positive ($ 11 > 0 $), so we will continue with division.
Step 5 :
Divide $ 17 $ by $ \color{blue}{ 11 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 6 :
Divide $ 11 $ by $ \color{blue}{ 6 } $ and get the remainder
The remainder is still positive ($ 5 > 0 $), so we will continue with division.
Step 7 :
Divide $ 6 $ by $ \color{blue}{ 5 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 8 :
Divide $ 5 $ by $ \color{blue}{ 1 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 1 }} $.
We can summarize an algorithm into a following table.
| 457 | : | 378 | = | 1 | remainder ( 79 ) | ||||||||||||||
| 378 | : | 79 | = | 4 | remainder ( 62 ) | ||||||||||||||
| 79 | : | 62 | = | 1 | remainder ( 17 ) | ||||||||||||||
| 62 | : | 17 | = | 3 | remainder ( 11 ) | ||||||||||||||
| 17 | : | 11 | = | 1 | remainder ( 6 ) | ||||||||||||||
| 11 | : | 6 | = | 1 | remainder ( 5 ) | ||||||||||||||
| 6 | : | 5 | = | 1 | remainder ( 1 ) | ||||||||||||||
| 5 | : | 1 | = | 5 | remainder ( 0 ) | ||||||||||||||
| GCD = 1 | |||||||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.