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**aceOfPentacles** Does 4048x + 418y = 66 have integer solutions? If an integer solution exists, find the solution for which y is positive and small as possible.

- I can never seem to get these sorts of problems right...

Here's my working:

**GCD(4048, 418) = 22 and 22 | 66.**

So 4048*3 + 418*-29 = 22

which implies 4048*9 + 418*-87 = 66

so x = 9, y = -87

So solutions are:

x = 9 + 418t

y = -87 - 4048t

Now, y must be > 0 :

-87 - 4048t > 0

-4048t > 87

t < -87/4048

t < -0.02(2dp)

So t must be < -1 since we are dealing with real numbers.

When I try to go any further I get crazy results. (Am I doing it right so far?)

Any help would be greatly appreciated thanks.