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I am trying to get a practice problem solved and I basically came to the following result.

What I need to prove is that some $x$ should not be divisible by $y$. The form of $y$ is $k(s.p_1 + t.p_2)$ and $k$ does not divide $x$, $p_1$ does not divide $x$, and $p_2$ does not divide $x$.

Can we conclude from here that $x$ is not divisible by $y$?

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Hozikimaru
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  • This is hard to follow. Clearly, if $k$ isn't a divisor of $x$ then no multiple of $k$ could ever be a divisor of $x$. Is that really what you intended to ask? – lulu Feb 24 '22 at 18:08
  • @lulu, yes, that is pretty much what I was trying to ask but how could we prove that if k is not a divisor of x, then no multiples of k would be a divisor of x? – Hozikimaru Feb 24 '22 at 18:14
  • Just think about it. If $x=y\times N$ then $x=k\times ((sp_1+tp_2)\times N)$. – lulu Feb 24 '22 at 18:22
  • $y=kn\mid x\Rightarrow k\mid x,$ contra hypothesis, by "divides" is transitive (cf. dupe). – Bill Dubuque Feb 24 '22 at 18:45
  • I see, logically makes sense but since this is an academic exercise, I wanted to make sure that the answer fits to the academic standards. – Hozikimaru Feb 24 '22 at 18:47

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