Find Greatest Common Divisor of 45 and 125, using Euclidean algorithm.
Answer
The GCD of given numbers is 5.
Explanation
Step 1 :
Divide $ 125 $ by $ 45 $ and get the remainder
The remainder is positive ($ 35 > 0 $), so we will continue with division.
Step 2 :
Divide $ 45 $ by $ \color{blue}{ 35 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 3 :
Divide $ 35 $ 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.
| 125 | : | 45 | = | 2 | remainder ( 35 ) | ||||||
| 45 | : | 35 | = | 1 | remainder ( 10 ) | ||||||
| 35 | : | 10 | = | 3 | 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.