Find Greatest Common Divisor of 34 and 55, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 55 $ by $ 34 $ and get the remainder
The remainder is positive ($ 21 > 0 $), so we will continue with division.
Step 2 :
Divide $ 34 $ by $ \color{blue}{ 21 } $ and get the remainder
The remainder is still positive ($ 13 > 0 $), so we will continue with division.
Step 3 :
Divide $ 21 $ by $ \color{blue}{ 13 } $ and get the remainder
The remainder is still positive ($ 8 > 0 $), so we will continue with division.
Step 4 :
Divide $ 13 $ by $ \color{blue}{ 8 } $ and get the remainder
The remainder is still positive ($ 5 > 0 $), so we will continue with division.
Step 5 :
Divide $ 8 $ by $ \color{blue}{ 5 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 6 :
Divide $ 5 $ by $ \color{blue}{ 3 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 7 :
Divide $ 3 $ by $ \color{blue}{ 2 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 8 :
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.
| 55 | : | 34 | = | 1 | remainder ( 21 ) | ||||||||||||||
| 34 | : | 21 | = | 1 | remainder ( 13 ) | ||||||||||||||
| 21 | : | 13 | = | 1 | remainder ( 8 ) | ||||||||||||||
| 13 | : | 8 | = | 1 | remainder ( 5 ) | ||||||||||||||
| 8 | : | 5 | = | 1 | remainder ( 3 ) | ||||||||||||||
| 5 | : | 3 | = | 1 | remainder ( 2 ) | ||||||||||||||
| 3 | : | 2 | = | 1 | 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.