Determine all the integer solutions to $23x + 39y = 2$

Question

I would like to calculate all the solutions to this equation using Euclides' algorithm and linear combination after finding the GCD. I suppose it's easy, but I'm a beginner.

$23x + 39y = 2$

Answer 1

Using the euclidean algorithm:

$$23x + 39y = 2$$

We have to find the $\gcd(23,39)$

Therefore,

$$39 = 23*(1) + 16$$ $$23 = 16*(1) + 7$$ $$16 = 7*(2) + 2$$ $$7 = 2*(3) + 1$$ $$2 = 1*(2)$$

so the gcd is $1$.

Now we rewrite:

$$1 = 7 - 3*(2)$$ $$1 = 7-3*(16-7*(2))$$ $$1 = 7*(7) - 3*(16)$$ $$1 = 7*(23-16) - 3*(16)$$ $$1 = 7*(23) - 10*(16)$$ $$1 = 7*(23) - 10*(39-23)$$ $$1 = 17*(23) - 10*(39)$$ So we multiply by $2$, since we get our orignial equation now:

$$2 = 34*(23) - 20*(39)$$

So one solution can be $x = 34$, and $y = -20$

Now generalizing, we find that:

$$x = 34 + \frac{39}{1}n = 34 + 39n$$ $$y = -20 - \frac{23}{1}n = -20 - 23n$$

For all integers $n$.

Look at this for more help:

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