Find Greatest Common Divisor of 196 and 249, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 249 $ by $ 196 $ and get the remainder
The remainder is positive ($ 53 > 0 $), so we will continue with division.
Step 2 :
Divide $ 196 $ by $ \color{blue}{ 53 } $ and get the remainder
The remainder is still positive ($ 37 > 0 $), so we will continue with division.
Step 3 :
Divide $ 53 $ by $ \color{blue}{ 37 } $ and get the remainder
The remainder is still positive ($ 16 > 0 $), so we will continue with division.
Step 4 :
Divide $ 37 $ by $ \color{blue}{ 16 } $ and get the remainder
The remainder is still positive ($ 5 > 0 $), so we will continue with division.
Step 5 :
Divide $ 16 $ by $ \color{blue}{ 5 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 6 :
Divide $ 5 $ 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.
| 249 | : | 196 | = | 1 | remainder ( 53 ) | ||||||||||
| 196 | : | 53 | = | 3 | remainder ( 37 ) | ||||||||||
| 53 | : | 37 | = | 1 | remainder ( 16 ) | ||||||||||
| 37 | : | 16 | = | 2 | remainder ( 5 ) | ||||||||||
| 16 | : | 5 | = | 3 | remainder ( 1 ) | ||||||||||
| 5 | : | 1 | = | 5 | remainder ( 0 ) | ||||||||||
| GCD = 1 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.