-1

I'm reading trough a proof that the number of (group) homomorphisms $\mathbb{Z}_n\rightarrow \mathbb{Z}_m$ is $$\text{gcd}(n, m),$$ and this is the only step that I'm not understanding, namely, that the number of solutions to the equation $na\equiv _m0$ for $0\leq a <m$ is $\text{gcd}(n,m)$.

I would appreciate any help.

Bill Dubuque
  • 272,048
Sam
  • 4,734
  • Can you figure out which values of $a \in \Bbb Z$ will satisfy $na \equiv_m 0$? Try an example. I'd recommend $m = 8, n = 12$. – Ben Grossmann Jan 12 '20 at 20:08

1 Answers1

0

Write $n a \cong 0 \pmod{m}$ as $$ na = k m \text{.} $$ If $n$ and $m$ have a common factor, $d$, this reduces to $(n/d)a = k(m/d)$, equivalently a congruence with a smaller modulus $$ (n/d)a \cong 0 \pmod{m/d} $$ Then there is a solution in each of the $d$ copies of the interval of length $m/d$ in the interval of length $m$. (In other words, the $d$ copies are $[a]$, $[a]+m/d$, $[a]+2m/d$, $\dots$, $[a]+(d-1)m/d$, where $[a]$ is the least nonnegative residue congruent to $a$ modulo $m/d$.) Therefore, there are $d = \gcd(m,n)$ solutions.

Eric Towers
  • 67,037
  • 1
    \equiv, not \cong.... – Arturo Magidin Jan 12 '20 at 20:29
  • @ArturoMagidin : Modular congruences are congruences. – Eric Towers Jan 12 '20 at 20:30
  • 1
    Yes, but the usual symbol is $\equiv$ (\equiv), not $\cong$ (\cong)... – Arturo Magidin Jan 12 '20 at 20:31
  • @ArturoMagidin : Not for the first half of my life, and I will never support the wrong notation for congruences. – Eric Towers Jan 12 '20 at 20:32
  • Well, it will be a lesser part of your life as time goes on... this is more of a problem with the nomenclature Knuth chose for the mathematical symbols, not for the symbols themselves. The three lines has a much longer genealogy than the other symbol, which is usually used for “isomorphism” in algebra. But different strokes for different folks... – Arturo Magidin Jan 12 '20 at 20:34
  • 1
    @EricTowers Are you aware of anyone else that uses $,a\cong b,$ vs. the standard $,a\equiv b,$ for congruences in $,\Bbb Z?\ \ \ $ – Bill Dubuque Jan 12 '20 at 23:48
  • @BillDubuque : Yes. It was the standard notation in pretty much everything I read up to the 90s. This does roughly coincide with the transition from overstriking an equal sign with a tilde on a typewriter giving way to obviously wrong notation by TeX users, so Arturo's reference to Knuth is likely relevant. – Eric Towers Jan 13 '20 at 15:08
  • I've read thousands of papers in number theory and algebra in that period and earlier and I don't recall seeing such usage. I suspect it is quite rare. Do you have any links? The $\equiv$ notation goes back to the introduction of integer congruences by Gauss in Disq. Arith. where he wrote "Numerorum congruentiam hoc signo, $\equiv$, in posterum denotabimus, modulum ubi opus erit in clausulis adiungentes, $-16\equiv 9\ \ ({\rm mod}., 5),\ -7\equiv 15\ \ ({\rm mod}., 11)$", See here for the scanned page. – Bill Dubuque Jan 13 '20 at 15:38
  • An English translation is "Henceforth we shall designate congruence by the symbol $,\equiv$, joining to it in parentheses the modulus when necessary to do so; e.g. ..." – Bill Dubuque Jan 13 '20 at 22:23