Find Greatest Common Divisor of 12378 and 3054, using Euclidean algorithm.
Answer
The GCD of given numbers is 6.
Explanation
Step 1 :
Divide $ 12378 $ by $ 3054 $ and get the remainder
The remainder is positive ($ 162 > 0 $), so we will continue with division.
Step 2 :
Divide $ 3054 $ by $ \color{blue}{ 162 } $ and get the remainder
The remainder is still positive ($ 138 > 0 $), so we will continue with division.
Step 3 :
Divide $ 162 $ by $ \color{blue}{ 138 } $ and get the remainder
The remainder is still positive ($ 24 > 0 $), so we will continue with division.
Step 4 :
Divide $ 138 $ by $ \color{blue}{ 24 } $ and get the remainder
The remainder is still positive ($ 18 > 0 $), so we will continue with division.
Step 5 :
Divide $ 24 $ by $ \color{blue}{ 18 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 6 :
Divide $ 18 $ by $ \color{blue}{ 6 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 6 }} $.
We can summarize an algorithm into a following table.
| 12378 | : | 3054 | = | 4 | remainder ( 162 ) | ||||||||||
| 3054 | : | 162 | = | 18 | remainder ( 138 ) | ||||||||||
| 162 | : | 138 | = | 1 | remainder ( 24 ) | ||||||||||
| 138 | : | 24 | = | 5 | remainder ( 18 ) | ||||||||||
| 24 | : | 18 | = | 1 | remainder ( 6 ) | ||||||||||
| 18 | : | 6 | = | 3 | remainder ( 0 ) | ||||||||||
| GCD = 6 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.