Step by step
The question
Question: find gcd(252, 198) by Euclid's algorithm.
1 Divide the bigger by the smaller
Divide the bigger number by the smaller: 198 goes into 252 once, and 54 is left over.
Here is the trick.
A number that goes into 252 and 198 also goes into 54, because 54 is 252 minus 1 × 198.
So the gcd does not change, and the numbers get smaller.
2 Keep going
Now the same with 198 and 54: the divisor moves left, the remainder becomes the new divisor.
54 = 1 × 36 + 18.
36 = 2 × 18, and the remainder is 0.
We stop.
3 The last remainder before 0
Careful!
0 only tells us to stop.
The answer is the last remainder before it: 18.
The last remainder that is not 0 is 18.
So gcd(252, 198) = 18.
4 Check
18 goes into both: 252 ÷ 18 is 14, and 198 ÷ 18 is 11.
And 14 and 11 have no common factor, so nothing bigger than 18 goes into both.
Right!
Let's look back.
First: divide the bigger by the smaller.
Then: keep going.
Then: the last remainder before 0.
Last: check.