Question on the relationship of a quantity and its value modulo n

Question

Is it correct to say that if there are 13 chameleons, then there are $1\pmod 3$ chameleons?

If so, why?

Answer 1

Yes, sort of, and if you understand the statement is a one-directional implication.

$(1)\;$ There are $13$ chameleons, and $\quad 13 = 4\cdot 3 + 1 \equiv 1\pmod 3$.

$(2)\;$ Therefore, the number of chameleons, when divided by $3$, leaves a remainder of $1$, which I suspect is what the intended meaning of the notation used when concluding that there are $1\pmod 3$ chameleons.

Yes, $(1)\implies (2)$. The fact that there are 13 chameleons implies that there are $1 \pmod 3$ chameleons. So the statement you post is true.

But that is not to suggest that the converse is necessarily true. For example, pick $x \equiv 1 \pmod 3$. Then $x\in \{\cdots, -5, -2, 1, 4, 7, 10, 13, \cdots\}$. So knowing that there are $x$ chameleons such that $x \equiv 1 \pmod 3$ does NOT imply that $x$ has to be $13$.

The better way to summarize the statement is as follows:

If there are $13$ chameleons, then there are $13\equiv 1\pmod 3$ chameleons.