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Given $m\geq 2$, denote by $b^{-1}$ the inverse of $b\pmod{m}$. That is, $b^{-1}$ is the residue for which $bb^{-1}\equiv 1\pmod{m}$. Sadie wonders if $(a+b)^{-1}$ is always congruent to $a^{-1}+b^{-1}$ (modulo $m$). She tries the example $a=2$, $b=3$, and $m=11$. Let $L$ be the residue of $(2+3)^{-1}\pmod{11}$, and let $R$ be the residue of $2^{-1}+3^{-1}\pmod{11}$, where $L$ and $R$ are integers from $0$ to $10$ (inclusive). Find $L-R$.

 Oct 3, 2023

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