I really can't get my head around this "modulo" thing.
Can someone show me a general step-by-step procedure on how I would be able to find out the 5 modulo 10, or 10 modulo 5.
Also, what does this mean: 1/17 = 113 modulo 120 ?
Because when I calculate(using a calculator) 113 modulo 120, the result is 113. But what is the 1/17 standing for then?
THANK YOU!
Best Answer
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 5is0because the remainder of10 / 5is0.The result of
7 modulo 5is2because the remainder of7 / 5is2.The reason your calculator says
113 modulo 120 = 113is 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 $120$.