Find Greatest Common Divisor of 800 and 234, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 800 $ by $ 234 $ and get the remainder
The remainder is positive ($ 98 > 0 $), so we will continue with division.
Step 2 :
Divide $ 234 $ by $ \color{blue}{ 98 } $ and get the remainder
The remainder is still positive ($ 38 > 0 $), so we will continue with division.
Step 3 :
Divide $ 98 $ by $ \color{blue}{ 38 } $ and get the remainder
The remainder is still positive ($ 22 > 0 $), so we will continue with division.
Step 4 :
Divide $ 38 $ by $ \color{blue}{ 22 } $ and get the remainder
The remainder is still positive ($ 16 > 0 $), so we will continue with division.
Step 5 :
Divide $ 22 $ by $ \color{blue}{ 16 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 6 :
Divide $ 16 $ by $ \color{blue}{ 6 } $ and get the remainder
The remainder is still positive ($ 4 > 0 $), so we will continue with division.
Step 7 :
Divide $ 6 $ by $ \color{blue}{ 4 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 8 :
Divide $ 4 $ by $ \color{blue}{ 2 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 2 }} $.
We can summarize an algorithm into a following table.
| 800 | : | 234 | = | 3 | remainder ( 98 ) | ||||||||||||||
| 234 | : | 98 | = | 2 | remainder ( 38 ) | ||||||||||||||
| 98 | : | 38 | = | 2 | remainder ( 22 ) | ||||||||||||||
| 38 | : | 22 | = | 1 | remainder ( 16 ) | ||||||||||||||
| 22 | : | 16 | = | 1 | remainder ( 6 ) | ||||||||||||||
| 16 | : | 6 | = | 2 | remainder ( 4 ) | ||||||||||||||
| 6 | : | 4 | = | 1 | remainder ( 2 ) | ||||||||||||||
| 4 | : | 2 | = | 2 | remainder ( 0 ) | ||||||||||||||
| GCD = 2 | |||||||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.