Find Greatest Common Divisor of 123456 and 789, using Euclidean algorithm.
Answer
The GCD of given numbers is 3.
Explanation
Step 1 :
Divide $ 123456 $ by $ 789 $ and get the remainder
The remainder is positive ($ 372 > 0 $), so we will continue with division.
Step 2 :
Divide $ 789 $ by $ \color{blue}{ 372 } $ and get the remainder
The remainder is still positive ($ 45 > 0 $), so we will continue with division.
Step 3 :
Divide $ 372 $ by $ \color{blue}{ 45 } $ and get the remainder
The remainder is still positive ($ 12 > 0 $), so we will continue with division.
Step 4 :
Divide $ 45 $ by $ \color{blue}{ 12 } $ and get the remainder
The remainder is still positive ($ 9 > 0 $), so we will continue with division.
Step 5 :
Divide $ 12 $ by $ \color{blue}{ 9 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 6 :
Divide $ 9 $ 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.
| 123456 | : | 789 | = | 156 | remainder ( 372 ) | ||||||||||
| 789 | : | 372 | = | 2 | remainder ( 45 ) | ||||||||||
| 372 | : | 45 | = | 8 | remainder ( 12 ) | ||||||||||
| 45 | : | 12 | = | 3 | remainder ( 9 ) | ||||||||||
| 12 | : | 9 | = | 1 | remainder ( 3 ) | ||||||||||
| 9 | : | 3 | = | 3 | remainder ( 0 ) | ||||||||||
| GCD = 3 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.