# What is the meaning of this Modulus output?

Quick question, what is the meaning of the $$\mathbb{c}_1$$ subscript in the output of Modulus (shown as an image below)?

I understand there are two solutions to this linear congruence, 6 and 15 (6+9), so I can guess $$\mathbb{c}_1$$ might denote something related to that.

• C[1] denotes an arbitrary (integer) constant. There are not only two solutions! There are infinitely many solutions, such as $x\in \{..., -21, -12, -3, 6, 15, 24, 33, ...\}$. Oct 29, 2021 at 13:51
• @Domen Thank you! Oct 29, 2021 at 14:01
• @Domen We are working modulo 18 though, which collapses this infinite set into $\{6, 15\}$. Oct 29, 2021 at 14:24
• @Szabolcs, ah, I see! I mistakenly thought this was equivalent to $4x=16 \; (\text{mod }18)$, but the option Modulus in fact defines a modular domain of solutions. Perhaps a useful note to the author: use Solve to get two solutions Solve[4 x == 6, x, Modulus -> 18] (* {{x -> 6}, {x -> 15}} *). Oct 29, 2021 at 14:42
• @Domen I think that deserves an answer (the use of Solve). Oct 29, 2021 at 15:02

When C[i] appears in the result, it usually means that the solution is valid for any value of C[i].
In this case, 6 + 9 c is a solution for any c. If you put in all possible values of c, you will see that 6 + 9 c is either 6 or 15:
Union@Table[Mod[6 + 9 c, 18], {c, 0, 17}]