Find Greatest Common Divisor of 1235 and 765, using Euclidean algorithm.
Answer
The GCD of given numbers is 5.
Explanation
Step 1 :
Divide $ 1235 $ by $ 765 $ and get the remainder
The remainder is positive ($ 470 > 0 $), so we will continue with division.
Step 2 :
Divide $ 765 $ by $ \color{blue}{ 470 } $ and get the remainder
The remainder is still positive ($ 295 > 0 $), so we will continue with division.
Step 3 :
Divide $ 470 $ by $ \color{blue}{ 295 } $ and get the remainder
The remainder is still positive ($ 175 > 0 $), so we will continue with division.
Step 4 :
Divide $ 295 $ by $ \color{blue}{ 175 } $ and get the remainder
The remainder is still positive ($ 120 > 0 $), so we will continue with division.
Step 5 :
Divide $ 175 $ by $ \color{blue}{ 120 } $ and get the remainder
The remainder is still positive ($ 55 > 0 $), so we will continue with division.
Step 6 :
Divide $ 120 $ by $ \color{blue}{ 55 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 7 :
Divide $ 55 $ by $ \color{blue}{ 10 } $ and get the remainder
The remainder is still positive ($ 5 > 0 $), so we will continue with division.
Step 8 :
Divide $ 10 $ by $ \color{blue}{ 5 } $ and get the remainder
The remainder is zero => GCD is the last divisor $ \color{blue}{ \boxed { 5 }} $.
We can summarize an algorithm into a following table.
| 1235 | : | 765 | = | 1 | remainder ( 470 ) | ||||||||||||||
| 765 | : | 470 | = | 1 | remainder ( 295 ) | ||||||||||||||
| 470 | : | 295 | = | 1 | remainder ( 175 ) | ||||||||||||||
| 295 | : | 175 | = | 1 | remainder ( 120 ) | ||||||||||||||
| 175 | : | 120 | = | 1 | remainder ( 55 ) | ||||||||||||||
| 120 | : | 55 | = | 2 | remainder ( 10 ) | ||||||||||||||
| 55 | : | 10 | = | 5 | remainder ( 5 ) | ||||||||||||||
| 10 | : | 5 | = | 2 | remainder ( 0 ) | ||||||||||||||
| GCD = 5 | |||||||||||||||||||
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.