Find Greatest Common Divisor of 83 and 49, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 83 $ by $ 49 $ and get the remainder
The remainder is positive ($ 34 > 0 $), so we will continue with division.
Step 2 :
Divide $ 49 $ by $ \color{blue}{ 34 } $ and get the remainder
The remainder is still positive ($ 15 > 0 $), so we will continue with division.
Step 3 :
Divide $ 34 $ by $ \color{blue}{ 15 } $ and get the remainder
The remainder is still positive ($ 4 > 0 $), so we will continue with division.
Step 4 :
Divide $ 15 $ by $ \color{blue}{ 4 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 5 :
Divide $ 4 $ by $ \color{blue}{ 3 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 6 :
Divide $ 3 $ 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.
| 83 | : | 49 | = | 1 | remainder ( 34 ) | ||||||||||
| 49 | : | 34 | = | 1 | remainder ( 15 ) | ||||||||||
| 34 | : | 15 | = | 2 | remainder ( 4 ) | ||||||||||
| 15 | : | 4 | = | 3 | remainder ( 3 ) | ||||||||||
| 4 | : | 3 | = | 1 | remainder ( 1 ) | ||||||||||
| 3 | : | 1 | = | 3 | remainder ( 0 ) | ||||||||||
| GCD = 1 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.