<s>[INST] Finding all elements in $x\in U_{143}$ such that $x^2=1 \pmod{143}$ 
Find all elements in $x\in U_{143}$ such that $x^2=1 \pmod{143}$

I am kind of stuck here. I know that $143=11*13$, and perhaps looking at $U_{11},U_{13}$ will help but I am unable to find a solution so far.
 [/INST] So we have $13\mid (x-1)(x+1)$ so $x=13k\pm 1$ and similary $x=11l\pm 1$


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*if $13k+1 = 11l+1\implies 11\mid k\implies k=11s \implies x= 143s+1\implies \boxed{x=1}$ 

*if $13k-1 = 11l-1\implies 11\mid k\implies k=11s \implies $$x= 143s-1\implies \boxed{ x=142}$ 

*if $13k+1 = 11l-1\implies 11\mid 13k+2\implies 11\mid 2(k+1) \implies $ $k=11s-1\implies x=143s-12 \implies $ $ 
\boxed{x= 131}$ 

*if $13k-1 = 11l+1\implies 11\mid 13k-2\implies 11\mid 2(k-1) \implies $ $k=11s+1\implies x=143s+12 \implies \boxed{x= 12}$ 
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