Find Greatest Common Divisor of 123 and 456, using Euclidean algorithm.
Answer
The GCD of given numbers is 3.
Explanation
Step 1 :
Divide $ 456 $ by $ 123 $ and get the remainder
The remainder is positive ($ 87 > 0 $), so we will continue with division.
Step 2 :
Divide $ 123 $ by $ \color{blue}{ 87 } $ and get the remainder
The remainder is still positive ($ 36 > 0 $), so we will continue with division.
Step 3 :
Divide $ 87 $ by $ \color{blue}{ 36 } $ and get the remainder
The remainder is still positive ($ 15 > 0 $), so we will continue with division.
Step 4 :
Divide $ 36 $ by $ \color{blue}{ 15 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 5 :
Divide $ 15 $ by $ \color{blue}{ 6 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 6 :
Divide $ 6 $ 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.
| 456 | : | 123 | = | 3 | remainder ( 87 ) | ||||||||||
| 123 | : | 87 | = | 1 | remainder ( 36 ) | ||||||||||
| 87 | : | 36 | = | 2 | remainder ( 15 ) | ||||||||||
| 36 | : | 15 | = | 2 | remainder ( 6 ) | ||||||||||
| 15 | : | 6 | = | 2 | remainder ( 3 ) | ||||||||||
| 6 | : | 3 | = | 2 | remainder ( 0 ) | ||||||||||
| GCD = 3 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.