Find Greatest Common Divisor of 25 and 68, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 68 $ by $ 25 $ and get the remainder
The remainder is positive ($ 18 > 0 $), so we will continue with division.
Step 2 :
Divide $ 25 $ by $ \color{blue}{ 18 } $ and get the remainder
The remainder is still positive ($ 7 > 0 $), so we will continue with division.
Step 3 :
Divide $ 18 $ by $ \color{blue}{ 7 } $ and get the remainder
The remainder is still positive ($ 4 > 0 $), so we will continue with division.
Step 4 :
Divide $ 7 $ 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.
| 68 | : | 25 | = | 2 | remainder ( 18 ) | ||||||||||
| 25 | : | 18 | = | 1 | remainder ( 7 ) | ||||||||||
| 18 | : | 7 | = | 2 | remainder ( 4 ) | ||||||||||
| 7 | : | 4 | = | 1 | 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.