What does $5\mathbb{Z}_{25}$ mean? (Notation help)

Question

Does $5 \mathbb{Z}_{25}$ refer to the numbers in $\mathbb{Z}_{25}$ multiplied by $5 $ -- i.e., there are 25 elements -- or does it refer to multiples of $5$ in $\mathbb{Z}_{25}$ -- i.e., there are $5$ elements? I can't figure out how to search for what this means, and it was never explained in class.

Answer 1

Both ways of thinking about $5\mathbb Z_{25}$ are essentially the same. Modulo $25$, $$5\times0\equiv5\times5\equiv5\times10\equiv5\times15\equiv5\times20\equiv0,$$ $$5\times1\equiv5\times6\equiv5\times11\equiv5\times16\equiv5\times21\equiv5,$$ $$5\times2\equiv5\times7\equiv5\times12\equiv5\times17\equiv5\times22\equiv10, $$ $$5\times3\equiv5\times8\equiv5\times13\equiv5\times18\equiv5\times23\equiv15,$$ and $$5\times4\equiv5\times9\equiv5\times14\equiv5\times19\equiv5\times24\equiv20,$$ so the elements of $\mathbb Z_{25}$ multiplied by $5$ are represented by the residue classes of the multiples of $5$;

there are $5$ elements.

$5\mathbb Z_{25}$ can also be thought of as the image of $5\mathbb Z$ under the homomorphism from $\mathbb Z$ to $\mathbb Z_{25}.$