Find Greatest Common Divisor of 23 and 85, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 85 $ by $ 23 $ and get the remainder
The remainder is positive ($ 16 > 0 $), so we will continue with division.
Step 2 :
Divide $ 23 $ by $ \color{blue}{ 16 } $ and get the remainder
The remainder is still positive ($ 7 > 0 $), so we will continue with division.
Step 3 :
Divide $ 16 $ by $ \color{blue}{ 7 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 4 :
Divide $ 7 $ by $ \color{blue}{ 2 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 5 :
Divide $ 2 $ 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.
| 85 | : | 23 | = | 3 | remainder ( 16 ) | ||||||||
| 23 | : | 16 | = | 1 | remainder ( 7 ) | ||||||||
| 16 | : | 7 | = | 2 | remainder ( 2 ) | ||||||||
| 7 | : | 2 | = | 3 | remainder ( 1 ) | ||||||||
| 2 | : | 1 | = | 2 | remainder ( 0 ) | ||||||||
| GCD = 1 | |||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.