2
$\begingroup$

If we divide the integers into groups of 3 ("ternary groups"), beginning with 3, we have group one $= 3,4,5$; group two $= 6,7,8$; and so on. If we ask which multiples of 5 belong to which groups, the answer is: 5 in group one, none in group two, 10 in group three, none in group four, and 15 in group five. So we have this pattern: $2\ \_\ 1\ \_\ 0$, where each value is the multiple of 5 modulo 3, and each index of the pattern corresponds to a ternary group. There are 5 places in the pattern, so the length of one cycle period is $5$. The blank value ("_") means no multiple of 5 is in the corresponding group. This pattern repeats for the next three multiples of 5, then the next three, on to infinity. The same applies to all values of $n$, e.g. the pattern for 7 is $\_\ 1\ \_\ 2\ \_\ \_\ 0$. The cycle is always of length $n$, and always ends with $3n \pmod 3$ (meaning it always end with 0, which corresponds to the third multiple of $n$). My question is, what is the formal rubric for this cycle? Where should I look to learn more about this phenomenon?

$\endgroup$
2
$\begingroup$

This is an example of the Chinese Remainder Theorem. If you have coprime moduli $a$ and $b$ you can solve the equations $$x \equiv c \pmod a\\ x \equiv d \pmod b$$ for any given $c,d$. The solutions will recur at intervals of $ab$. In your example when you note that $5$ is the middle of a group of $3$ you are solving $$x\equiv 2 \pmod 3 \\ x\equiv 0 \pmod 5$$ As you noticed the first solution is $x=5$ and the next is $15$ higher or $20$ and so on.

$\endgroup$
1
  • $\begingroup$ Thank you, this is greatly appreciated. $\endgroup$ – In Lak'ech Jul 29 '18 at 12:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.