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I'm solving this equation in integers: $6 \, x + 10 \, y + 15 \, z = 1$.

I can find many solutions, by combining two two-variable equations into $x = u \, p$, $y = v \, p$, $z = q$, where $u = 2 + 5 \, t$, $v = -1 - 3 \, t$, and $p = 8 + 15 \, s$, $q = -1 - 2 \, s$, with $s$ and $t$ integers. (see Wikipedia)

But I'm obviously missing some solutions. How does Mathematica arrive at this elegant general solution?

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  • $\begingroup$ @DietrichBurde Probably. I had searched for such an answer without success. $\endgroup$
    – BoLe
    Commented Nov 13, 2020 at 12:09
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    $\begingroup$ I added a link which gives the general method. $\endgroup$ Commented Nov 13, 2020 at 23:10
  • $\begingroup$ @BillDubuque Thanks. Second link proves especially useful, as Adam Bailey there suggests simple modular arithmetic which readily leads to a desired general solution. $\endgroup$
    – BoLe
    Commented Nov 14, 2020 at 12:35
  • $\begingroup$ The remark you refer to is a part of the general (recursive) algorithm that I describe there (which works for any number of variables). But the algorithm involves (much) more than that simple remark, so I highly suggest that you learn the (ideas behind) the general algorithm if you wish to master such. $\endgroup$ Commented Nov 14, 2020 at 20:51

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