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For any integer x, if x mod 15=7 then x mod 5=2,

Make an outline for a proof of this statement being as explicit as possible, then prove the statement.

I am a bit stuck on how to transform this into symbols. So far I have this: ∀x∈Z(x mod 15=7 → ∃q∈Z...?)

JMoravitz
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OliOlivee
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  • If you wish to ask a question about logic (vs. number theory) then you should pose another question highlighting the logical aspects that trouble you and tag it appropriately. – Bill Dubuque Oct 22 '20 at 00:21
  • @BillDubuque Thank you, I was not aware that number theory was a topic. The class I am taking is called Sets and Logic, so I assumed that everything would be in logic. – OliOlivee Oct 22 '20 at 00:38
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    Oh, I see now that the your tags were changed by another user. At this point probably the best thing to do if you still have logical doubts is to pose a new question and make it much clearer in the question precisely what logical difficulties you have. – Bill Dubuque Oct 22 '20 at 00:44

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For integers $x,y$ and positive integer $n$ you have by definition that $x\equiv y\pmod{n}$ is true if and only if there exists an integer $k$ such that $x=y+nk$

Here, $x\equiv 7\pmod{15}$ means that there exists some integer $k$ such that $x=15k+7$

We want to show that $x\equiv 2\pmod{5}$ and we would be able to show this by showing that $x=5\ell + 2$ with $\ell$ some integer. Find that integer $\ell$.

$x=15k+7\implies x=5\cdot 3k + 5 + 2\implies x = 5(3k+1)+2$

JMoravitz
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