Find Greatest Common Divisor of 12345 and 54321, using Euclidean algorithm.
Answer
The GCD of given numbers is 3.
Explanation
Step 1 :
Divide $ 54321 $ by $ 12345 $ and get the remainder
The remainder is positive ($ 4941 > 0 $), so we will continue with division.
Step 2 :
Divide $ 12345 $ by $ \color{blue}{ 4941 } $ and get the remainder
The remainder is still positive ($ 2463 > 0 $), so we will continue with division.
Step 3 :
Divide $ 4941 $ by $ \color{blue}{ 2463 } $ and get the remainder
The remainder is still positive ($ 15 > 0 $), so we will continue with division.
Step 4 :
Divide $ 2463 $ by $ \color{blue}{ 15 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 5 :
Divide $ 15 $ by $ \color{blue}{ 3 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 3 }} $.
We can summarize an algorithm into a following table.
| 54321 | : | 12345 | = | 4 | remainder ( 4941 ) | ||||||||
| 12345 | : | 4941 | = | 2 | remainder ( 2463 ) | ||||||||
| 4941 | : | 2463 | = | 2 | remainder ( 15 ) | ||||||||
| 2463 | : | 15 | = | 164 | remainder ( 3 ) | ||||||||
| 15 | : | 3 | = | 5 | remainder ( 0 ) | ||||||||
| GCD = 3 | |||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.