Find Greatest Common Divisor of 78 and 14, using Euclidean algorithm.
Answer
The GCD of given numbers is 2.
Explanation
Step 1 :
Divide $ 78 $ by $ 14 $ and get the remainder
The remainder is positive ($ 8 > 0 $), so we will continue with division.
Step 2 :
Divide $ 14 $ by $ \color{blue}{ 8 } $ and get the remainder
The remainder is still positive ($ 6 > 0 $), so we will continue with division.
Step 3 :
Divide $ 8 $ by $ \color{blue}{ 6 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 4 :
Divide $ 6 $ 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 | : | 14 | = | 5 | remainder ( 8 ) | ||||||
| 14 | : | 8 | = | 1 | remainder ( 6 ) | ||||||
| 8 | : | 6 | = | 1 | remainder ( 2 ) | ||||||
| 6 | : | 2 | = | 3 | remainder ( 0 ) | ||||||
| GCD = 2 | |||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.