Find Greatest Common Divisor of 78 and 32, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 78 $ by $ 32 $ and get the remainder
The remainder is positive ($ 14 > 0 $), so we will continue with division.
Step 2 :
Divide $ 32 $ by $ \color{blue}{ 14 } $ and get the remainder
The remainder is still positive ($ 4 > 0 $), so we will continue with division.
Step 3 :
Divide $ 14 $ by $ \color{blue}{ 4 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 4 :
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.
| 78 | : | 32 | = | 2 | remainder ( 14 ) | ||||||
| 32 | : | 14 | = | 2 | remainder ( 4 ) | ||||||
| 14 | : | 4 | = | 3 | 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.