Find Greatest Common Divisor of 48 and 31, using Euclidean algorithm.
Answer
The GCD of given numbers is 1.
Explanation
Step 1 :
Divide $ 48 $ by $ 31 $ and get the remainder
The remainder is positive ($ 17 > 0 $), so we will continue with division.
Step 2 :
Divide $ 31 $ by $ \color{blue}{ 17 } $ and get the remainder
The remainder is still positive ($ 14 > 0 $), so we will continue with division.
Step 3 :
Divide $ 17 $ by $ \color{blue}{ 14 } $ and get the remainder
The remainder is still positive ($ 3 > 0 $), so we will continue with division.
Step 4 :
Divide $ 14 $ by $ \color{blue}{ 3 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 5 :
Divide $ 3 $ by $ \color{blue}{ 2 } $ and get the remainder
The remainder is still positive ($ 1 > 0 $), so we will continue with division.
Step 6 :
Divide $ 2 $ by $ \color{blue}{ 1 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 1 }} $.
We can summarize an algorithm into a following table.
| 48 | : | 31 | = | 1 | remainder ( 17 ) | ||||||||||
| 31 | : | 17 | = | 1 | remainder ( 14 ) | ||||||||||
| 17 | : | 14 | = | 1 | remainder ( 3 ) | ||||||||||
| 14 | : | 3 | = | 4 | remainder ( 2 ) | ||||||||||
| 3 | : | 2 | = | 1 | remainder ( 1 ) | ||||||||||
| 2 | : | 1 | = | 2 | remainder ( 0 ) | ||||||||||
| GCD = 1 | |||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.