The GCD of given numbers is 2.
Step 1 :
Divide $ 2378 $ by $ 1764 $ and get the remainder
The remainder is positive ($ 614 > 0 $), so we will continue with division.
Step 2 :
Divide $ 1764 $ by $ \color{blue}{ 614 } $ and get the remainder
The remainder is still positive ($ 536 > 0 $), so we will continue with division.
Step 3 :
Divide $ 614 $ by $ \color{blue}{ 536 } $ and get the remainder
The remainder is still positive ($ 78 > 0 $), so we will continue with division.
Step 4 :
Divide $ 536 $ by $ \color{blue}{ 78 } $ and get the remainder
The remainder is still positive ($ 68 > 0 $), so we will continue with division.
Step 5 :
Divide $ 78 $ by $ \color{blue}{ 68 } $ and get the remainder
The remainder is still positive ($ 10 > 0 $), so we will continue with division.
Step 6 :
Divide $ 68 $ by $ \color{blue}{ 10 } $ and get the remainder
The remainder is still positive ($ 8 > 0 $), so we will continue with division.
Step 7 :
Divide $ 10 $ by $ \color{blue}{ 8 } $ and get the remainder
The remainder is still positive ($ 2 > 0 $), so we will continue with division.
Step 8 :
Divide $ 8 $ 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.
2378 | : | 1764 | = | 1 | remainder ( 614 ) | ||||||||||||||
1764 | : | 614 | = | 2 | remainder ( 536 ) | ||||||||||||||
614 | : | 536 | = | 1 | remainder ( 78 ) | ||||||||||||||
536 | : | 78 | = | 6 | remainder ( 68 ) | ||||||||||||||
78 | : | 68 | = | 1 | remainder ( 10 ) | ||||||||||||||
68 | : | 10 | = | 6 | remainder ( 8 ) | ||||||||||||||
10 | : | 8 | = | 1 | remainder ( 2 ) | ||||||||||||||
8 | : | 2 | = | 4 | remainder ( 0 ) | ||||||||||||||
GCD = 2 |
This solution can be visualized using a Venn diagram.
The GCD equals the product of the numbers at the intersection.