Hi everyone!

I have the following problem:

Which of these equations have an integer solution:

3731s + 6149t = 26

1001s + 2261t = 1

495s + 1704t = 1

385s + 17081t = 1

I thought the first one would not have an integer solution because 26 is not the gcd of 3731 and 6149, but 13 is. So I thought that it does not have an integer solution because if it did the equation would have to be 3731s + 6149t = 13, but according to the answer key it does have an integer solution. Could anyone explain why?

Thank you everyone!