Find Greatest Common Divisor of 2689 and 4001, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 4001 $ by $ 2689 $ and get the remainder
The remainder is positive ($ 1312 > 0 $), so we will continue with division.
Step 2 :
Divide $ 2689 $ by $ \color{blue}{ 1312 } $ and get the remainder
The remainder is still positive ($ 65 > 0 $), so we will continue with division.
Step 3 :
Divide $ 1312 $ by $ \color{blue}{ 65 } $ and get the remainder
The remainder is still positive ($ 12 > 0 $), so we will continue with division.
Step 4 :
Divide $ 65 $ by $ \color{blue}{ 12 } $ and get the remainder
The remainder is still positive ($ 5 > 0 $), so we will continue with division.
Step 5 :
Divide $ 12 $ by $ \color{blue}{ 5 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 6 :
Divide $ 5 $ by $ \color{blue}{ 2 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 7 :
Divide $ 2 $ 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.
| 4001 | : | 2689 | = | 1 | remainder ( 1312 ) | ||||||||||||
| 2689 | : | 1312 | = | 2 | remainder ( 65 ) | ||||||||||||
| 1312 | : | 65 | = | 20 | remainder ( 12 ) | ||||||||||||
| 65 | : | 12 | = | 5 | remainder ( 5 ) | ||||||||||||
| 12 | : | 5 | = | 2 | remainder ( 2 ) | ||||||||||||
| 5 | : | 2 | = | 2 | remainder ( 1 ) | ||||||||||||
| 2 | : | 1 | = | 2 | remainder ( 0 ) | ||||||||||||
| GCD = 1 | |||||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.