Step by step
The question
Question: find gcd(270, 192) by Euclid's algorithm.
1 Divide the bigger by the smaller
Divide the bigger number by the smaller: 192 goes into 270 once, and 78 is left over.
Here is the trick.
A number that goes into 270 and 192 also goes into 78, because 78 is 270 minus 1 × 192.
So the gcd does not change, and the numbers get smaller.
2 Keep going
Now the same with 192 and 78: the divisor moves left, the remainder becomes the new divisor.
78 = 2 × 36 + 6.
36 = 6 × 6, 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: 6.
The last remainder that is not 0 is 6.
So gcd(270, 192) = 6.
4 Check
6 goes into both: 270 ÷ 6 is 45, and 192 ÷ 6 is 32.
And 45 and 32 have no common factor, so nothing bigger than 6 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.