Find Greatest Common Divisor of 69 and 372, using Euclidean algorithm.
Answer
The GCD of given numbers is 3.
Explanation
Step 1 :
Divide $ 372 $ by $ 69 $ and get the remainder
The remainder is positive ($ 27 > 0 $), so we will continue with division.
Step 2 :
Divide $ 69 $ by $ \color{blue}{ 27 } $ and get the remainder
The remainder is still positive ($ 15 > 0 $), so we will continue with division.
Step 3 :
Divide $ 27 $ by $ \color{blue}{ 15 } $ and get the remainder
The remainder is still positive ($ 12 > 0 $), so we will continue with division.
Step 4 :
Divide $ 15 $ by $ \color{blue}{ 12 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 5 :
Divide $ 12 $ 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.
| 372 | : | 69 | = | 5 | remainder ( 27 ) | ||||||||
| 69 | : | 27 | = | 2 | remainder ( 15 ) | ||||||||
| 27 | : | 15 | = | 1 | remainder ( 12 ) | ||||||||
| 15 | : | 12 | = | 1 | remainder ( 3 ) | ||||||||
| 12 | : | 3 | = | 4 | remainder ( 0 ) | ||||||||
| GCD = 3 | |||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.