Find Greatest Common Divisor of 54 and 21, using Euclidean algorithm.
Answer
The GCD of given numbers is 3.
Explanation
Step 1 :
Divide $ 54 $ by $ 21 $ and get the remainder
The remainder is positive ($ 12 > 0 $), so we will continue with division.
Step 2 :
Divide $ 21 $ by $ \color{blue}{ 12 } $ and get the remainder
The remainder is still positive ($ 9 > 0 $), so we will continue with division.
Step 3 :
Divide $ 12 $ by $ \color{blue}{ 9 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 4 :
Divide $ 9 $ by $ \color{blue}{ 3 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 3 }} $.
We can summarize an algorithm into a following table.
| 54 | : | 21 | = | 2 | remainder ( 12 ) | ||||||
| 21 | : | 12 | = | 1 | remainder ( 9 ) | ||||||
| 12 | : | 9 | = | 1 | remainder ( 3 ) | ||||||
| 9 | : | 3 | = | 3 | remainder ( 0 ) | ||||||
| GCD = 3 | |||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.