How to calculate a Modulo?

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!

Solution 1:

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 $120$.

Solution 2:

There are ways to calculate it, modulo is remainder counting basically. $$7 = 2 \mod 5$$ because $7=5*1+2$ $$12 = 2 \mod 5$$ because $12=5*2+2$ and so on, so if you want to calculate for example $73 = a \mod 7$ you can do this, that is want to get $a$, take 73 and continue subtracting 7 until you no longer can. $73-7=66$, $66-7=59$ etc until we get $10-7=3$ which gives us that $a=3$ in it's simplest form (any of the results along the way can technically be a).

As for what $1/17=113 \mod 120$ the question is simply what times 17 gives remainder 1 when divided by 120? $113\cdot 17 = 1921 = 120\cdot 16+1$

Solution 3:

The "mod" operator in computer languages is simply the remainder. For example,

17 mod 3 = 2

because

17 / 3 = 5 rem 2

which in turn means

17 = 3 * 5 + 2

Solution 4:

A simple way to understand this "modulo" operation. Consider the positive integers $7,12,17,22,27,32,...$ All these integers leave $5$ as the remainder when divided by $5$. In other words, if we subtract $2$ from each integer, the resulting set is a set multiples of $5$. We can travel on the other direction also. The integers $2,-3,-8,...$ also have that property, i.e., if we subtract $2$ from each integer, the resulting set is a set of multiples of $5$. Thus the set $\{...-8,-3,2,7,12,17,22,...\}$ will become a set of multiples of $5$ if we subtract $2$ from each. We call this set as $2$ modulo $5$. So, whenever we say $2$ modulo $5$, we mean a number which leaves $2$ as the remainder when divided by $5$. If $x \equiv 2 \pmod 5$, then $x -2$ is a multiple of $5$.