$\!\!\bmod 24\!:\ mn+1\equiv 0\Rightarrow m+n \equiv 0\,$ using group theory

Question

I would like to prove that : if $m, n \in \mathbb{N}$ such that : $mn+1 \equiv 0 \pmod{24}$ then $m+n \equiv 0 \pmod{24}$.

I know how to prove that but it's quite annoying : just look at every possible rest of $m$ and $n$ mod $24$. Yet it's annoying because they are $24^2$ cases to be treated.

I think we can solve that more easily using group theory, that's why I am just asking : does any one have an idea of how to solve this problem using group theory ? Maybe by looking at : $\mathbb{F}_{3}$ and $\mathbb{F}_2$...

Answer 1

You're trying to prove that if $mn \equiv -1 \pmod{24}$ then $m \equiv -n \pmod{24}$.

Let $k = -n$. Then you're trying to show that if $-mk \equiv -1 \pmod{24}$ then $m \equiv k \pmod{24}$.

Of course if $-mk \equiv -1$ then $mk \equiv 1$. So what you want to show is that in the integers $\mod 24$, every invertible element is its own inverse. Group-theoretically, this amounts to showing that every element of $(\mathbb{Z}/24\mathbb{Z})^\times $has order $1$ or $2$.

But by the Chinese remainder theorem, $(\mathbb{Z}/24\mathbb{Z})^\times$ is isomorphic to $(\mathbb{Z}/3\mathbb{Z})^\times \times (\mathbb{Z}/8\mathbb{Z})^\times$. It's enough to show that every non-identity element of $(\mathbb{Z}/3\mathbb{Z})^\times$ and of $(\mathbb{Z}/8\mathbb{Z})^\times$ has order $2$. But that's easy to do by direct computation: $2^2 \equiv 1 \pmod{3}$ and $3^2 \equiv 5^2 \equiv 7^2 \equiv 1 \pmod{8}$.

Answer 2

Swapping sign of $\:\!n\:\!$ it becomes $\,mn \equiv 1\Rightarrow m\equiv n,\,$ i.e. $\:\!n\:\!$ invertible $\:\!\Rightarrow\, n^{-1}\equiv n\, [\!\iff\! n^2\equiv 1],\,$ true mod $8$ and mod $3$, since invertibles mod $8$ are odd, therefore $\,{\rm odd}^2\!\equiv \{\pm1, \pm3\}^2\equiv 1,$ and mod $3$ invertibles are $\pm 1$ which also square to $1$. Thus $\,3,8\mid n^2\!-\!1\,\Rightarrow\,{\rm lcm}(3,8)\!=\! 24\mid n^2\!-\!1.\,$ Proving the converse is trickier, but it has a one line proof.

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Via group theory: using Carmichael's lambda function (universal group exponent) we compute $\lambda(8\cdot 3) = {\rm lcm}(\phi(8)/2,\phi(3)) = {\rm lcm}(2,2) = \color{#c00}2,\,$ so $\,(n,24)=1\,\Rightarrow\, n^{\color{#c00}{\large 2}}\equiv 1\pmod{\!24}$

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Remark $ $ This characteristic property of $24$ lies at the heart of many results - even in more advanced contexts, e.g. see Chebolu: What Is Special about the Divisors of 24? (2017), and Gannon: Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics. (2006), and see the prior question A high-powered explanation for $\exp U(n)=2\iff n\mid24$?

Answer 3

The OP can solve this without resorting to the brute force breakdown of $24^2$ cases; they can employ 'staring you in the face' group theory and then roll up their sleeves.

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With

$$mn+1 \equiv 0 \pmod{24}$$

we can write

$m \cdot (-n) \equiv 1 \pmod{24}$
$(-m) \cdot n \equiv 1 \pmod{24}$

and so

$m^{-1} \equiv -n \pmod{24}$
$n^{-1} \equiv -m \pmod{24}$

It is elementary to write

$$ \mathbb{Z}/24\mathbb{Z}^\times = \{1,5,7,11,13,17,19,23\}$$

and to solve for $m$ and $n$:

$$ (m,n) \in A \cup A^T$$

where

$$ A = \{(1,23),(5,19),(7,17),(11,13)\}$$

So as they say, "if the glove fits you must wear it" and, indeed,

$$\bmod 24\!:\ mn+1\equiv 0\Rightarrow m+n \equiv 0$$