Step by step
The question
Question: find gcd(210, 154) by Euclid's algorithm.
1 Divide the bigger by the smaller
Divide the bigger number by the smaller: 154 goes into 210 once, and 56 is left over.
Here is the trick.
A number that goes into 210 and 154 also goes into 56, because 56 is 210 minus 1 × 154.
So the gcd does not change, and the numbers get smaller.
2 Keep going
Now the same with 154 and 56: the divisor moves left, the remainder becomes the new divisor.
56 = 1 × 42 + 14.
42 = 3 × 14, 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: 14.
The last remainder that is not 0 is 14.
So gcd(210, 154) = 14.
4 Check
14 goes into both: 210 ÷ 14 is 15, and 154 ÷ 14 is 11.
And 15 and 11 have no common factor, so nothing bigger than 14 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.