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I am trying to prove that $22^{1/2}$ is irrational using the classic proof by contradiction. I need to prove an auxiliary modular lemma:

if $n \not\equiv 0$ (mod 22) then $n^{2} \not\equiv 0$ (mod 22).

I know I can just list out all 21 cases where the remainder is a number 1 through 21, but could I just use a smaller mod? Such as mod 2, or mod 11 because they are factors of 22? And somehow relate that back to mod 22?

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    "I need to prove an auxiliary modular lemma." There are faster ways to prove it. Do you need a proof that uses this? – 2017-02-27
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    @ajotatxe yes I have to use this – 2017-02-27
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    We can say $22$ divides $k$ if and only if $2$ divides $k$ and $k$ divides $k$. If you can prove that $2$ divides $n^2$ iff $2$ divides $n$, and the same for $11$, then you're good. – 2017-02-27
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    Do you already know that if a prime divides a product then it divides some factor of the product? – 2017-02-27

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Suppose $\;n^2=0\pmod{22}\;$ , then $\;n^2\;$ is even and divisible by $\;11\;$ , but then so is $\;n\;$ and there is the proof of your modular lemma (this is based on the fundamental theorem of arithmetic).

Now, suppose $\;\sqrt{22}=\frac mn\;$ , where $\;m,n\in\Bbb N\;,\;\;n\neq0\;$ is a reduced fraction , then:

$$22n^2\stackrel{(\color{red}*)}=m^2\overbrace{\implies}^{\text{the lemma}} m=0\pmod{22}\implies m=22 k\implies 22n^2\stackrel{(\color{red}*)}=22^2k^2\implies$$

$$n^2=22k^2\overbrace{\implies}^{\text{the lemma}} n=0\pmod{22}\;,\;\;\text{contradiction and we're done}$$