When you see "modulo", especially if you are using a calculator, think of it as the remainder term when you do division.

Examples:

The result of 10 modulo 5 is 0 because the remainder of 10 / 5 is 0.

The result of 7 modulo 5 is 2 because the remainder of 7 / 5 is 2.

The reason your calculator says 113 modulo 120 = 113 is because 113 < 120, so it isn't doing any division.

More generally, the idea is that two numbers are congruent if they are the same modulo a given number (or modulus)

For example, as above, $7 \equiv 2 \mod 5$ where $5$ is our modulus.

Another issue is that of inverses, which is where the confusion of $1/17$ comes in.

We say that $a$ and $b$ are inverses modulo $n$, if $ab \equiv 1 \mod n$, and we might write $b = a^{-1}$.

For example $17\cdot 113 = 1921 = 120\cdot 16 +1 \equiv 1 \mod 120$, so $17^{-1} = 113$ modulo $120$.