Find Greatest Common Divisor of 25 and 90, using Euclidean algorithm.
Answer
The GCD of given numbers is 5.
Explanation
Step 1 :
Divide $ 90 $ by $ 25 $ and get the remainder
The remainder is positive ($ 15 > 0 $), so we will continue with division.
Step 2 :
Divide $ 25 $ by $ \color{blue}{ 15 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 3 :
Divide $ 15 $ by $ \color{blue}{ 10 } $ and get the remainder
The remainder is still positive ($ 5 > 0 $), so we will continue with division.
Step 4 :
Divide $ 10 $ by $ \color{blue}{ 5 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 5 }} $.
We can summarize an algorithm into a following table.
| 90 | : | 25 | = | 3 | remainder ( 15 ) | ||||||
| 25 | : | 15 | = | 1 | remainder ( 10 ) | ||||||
| 15 | : | 10 | = | 1 | remainder ( 5 ) | ||||||
| 10 | : | 5 | = | 2 | remainder ( 0 ) | ||||||
| GCD = 5 | |||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.