Find Greatest Common Divisor of 232 and 562, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 562 $ by $ 232 $ and get the remainder
The remainder is positive ($ 98 > 0 $), so we will continue with division.
Step 2 :
Divide $ 232 $ by $ \color{blue}{ 98 } $ and get the remainder
The remainder is still positive ($ 36 > 0 $), so we will continue with division.
Step 3 :
Divide $ 98 $ by $ \color{blue}{ 36 } $ and get the remainder
The remainder is still positive ($ 26 > 0 $), so we will continue with division.
Step 4 :
Divide $ 36 $ by $ \color{blue}{ 26 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 5 :
Divide $ 26 $ by $ \color{blue}{ 10 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 6 :
Divide $ 10 $ 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.
| 562 | : | 232 | = | 2 | remainder ( 98 ) | ||||||||||||||
| 232 | : | 98 | = | 2 | remainder ( 36 ) | ||||||||||||||
| 98 | : | 36 | = | 2 | remainder ( 26 ) | ||||||||||||||
| 36 | : | 26 | = | 1 | remainder ( 10 ) | ||||||||||||||
| 26 | : | 10 | = | 2 | remainder ( 6 ) | ||||||||||||||
| 10 | : | 6 | = | 1 | 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.