Find Greatest Common Divisor of 117 and 189, using Euclidean algorithm.
Answer
The GCD of given numbers is 9.
Explanation
Step 1 :
Divide $ 189 $ by $ 117 $ and get the remainder
The remainder is positive ($ 72 > 0 $), so we will continue with division.
Step 2 :
Divide $ 117 $ by $ \color{blue}{ 72 } $ and get the remainder
The remainder is still positive ($ 45 > 0 $), so we will continue with division.
Step 3 :
Divide $ 72 $ by $ \color{blue}{ 45 } $ and get the remainder
The remainder is still positive ($ 27 > 0 $), so we will continue with division.
Step 4 :
Divide $ 45 $ by $ \color{blue}{ 27 } $ and get the remainder
The remainder is still positive ($ 18 > 0 $), so we will continue with division.
Step 5 :
Divide $ 27 $ by $ \color{blue}{ 18 } $ and get the remainder
The remainder is still positive ($ 9 > 0 $), so we will continue with division.
Step 6 :
Divide $ 18 $ by $ \color{blue}{ 9 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 9 }} $.
We can summarize an algorithm into a following table.
| 189 | : | 117 | = | 1 | remainder ( 72 ) | ||||||||||
| 117 | : | 72 | = | 1 | remainder ( 45 ) | ||||||||||
| 72 | : | 45 | = | 1 | remainder ( 27 ) | ||||||||||
| 45 | : | 27 | = | 1 | remainder ( 18 ) | ||||||||||
| 27 | : | 18 | = | 1 | remainder ( 9 ) | ||||||||||
| 18 | : | 9 | = | 2 | remainder ( 0 ) | ||||||||||
| GCD = 9 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.