Find Greatest Common Divisor of 1024 and 2256, using Euclidean algorithm.
Answer
The GCD of given numbers is 16.
Explanation
Step 1 :
Divide $ 2256 $ by $ 1024 $ and get the remainder
The remainder is positive ($ 208 > 0 $), so we will continue with division.
Step 2 :
Divide $ 1024 $ by $ \color{blue}{ 208 } $ and get the remainder
The remainder is still positive ($ 192 > 0 $), so we will continue with division.
Step 3 :
Divide $ 208 $ by $ \color{blue}{ 192 } $ and get the remainder
The remainder is still positive ($ 16 > 0 $), so we will continue with division.
Step 4 :
Divide $ 192 $ by $ \color{blue}{ 16 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 16 }} $.
We can summarize an algorithm into a following table.
| 2256 | : | 1024 | = | 2 | remainder ( 208 ) | ||||||
| 1024 | : | 208 | = | 4 | remainder ( 192 ) | ||||||
| 208 | : | 192 | = | 1 | remainder ( 16 ) | ||||||
| 192 | : | 16 | = | 12 | remainder ( 0 ) | ||||||
| GCD = 16 | |||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.