0

The question said:

Use the Euclidean Algorithm to find gcd $(1207,569)$ and write $(1207,569)$ as an integer linear combination of $1207$ and $569$

I proceeded as follows:

$$ 12007 = 569(2) +69$$

$$569 = 69(8) +17$$

$$69 = 17(4) +1$$

$$17 = 1(17) + 0$$

Thus the gcd = 1

The part I am having problems with is how calculate and write it was a linear combination. Can someone help?

David
  • 97
  • 8
  • 1
    Hint: Start substituting the LHS from the last equation to the RHS of each previous equation, up to the first equation. – John Omielan Mar 06 '19 at 23:55
  • If you look up "extended Euclid Algorithm" that shows how to do the Euclid algorithm along with about an equal amount of work on the side, so as to get both the gcd and a linear combination that adds to that gcd. – Mark Fischler Mar 07 '19 at 00:00
  • 1
    Please search first before posing questions. There are hundreds of posts on this topic. – Bill Dubuque Mar 07 '19 at 00:01
  • There is a direct algorithm, known as the extended Euclidean algorithm. You can find a description with justification in the answer to this recent question. – Bernard Mar 07 '19 at 00:16

1 Answers1

0

Now you need to go back in the steps. You know that $1=69-17\times 4$. From the previous step you know that $17=569-69\times 8$. Putting this to the previous equation we get:

$1=69-(569-69\times 8)\times 4$.

Finally if we take it one step back in the algorithm we get $69=1207-569\times 2$. So we get:

$1=1207-569\times 2-(569-(1207-569\times 2)\times 8)\times 4=1207\times 33+569\times (-70)$

Mark
  • 39,605