How much is 17 mod 3?
17 mod 3 equals 2 since dividing 17 by 3 gives a quotient of 5 and a remainder of 2. The remainder is the result of the modulus operation. In simpler terms, 17 mod 3 = 2.
How to calculate modulo division?
To calculate modulo division: subtract the divisor from the dividend until the resultant is less than the divisor.
What are the components of modulo division?
The components of modulo division are dividend, divisor, quotient, and remainder. The remainder is the answer or end result of the operation.
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The remainder when $1$ is divided by $2$ is $1$, since $1=(0)(2)+1$ and $0\le 1\lt 2$.
In general, if $0\le a\lt m$ then $a\operatorname{\%}m=a$.
In general, when you divide an integer $a$ by a positive integer $m$, there is a quotient $q$ and a remainder $r$. So $$a=qm+r,$$ where $0\le r\lt m$.
For instance, if $a=30$ and $m=12$, then $q=2$ and $r=6$. If $a=5$ and $m=12$, then $q=0$ and $r=5$.
In the case where $a=1$ and $m=2$, the quotient is $0$ and the remainder is $1$.
Remark: It is useful to have concrete images to go along with more abstract descriptions. Suppose that we have a box that contains $a$ cookies, and we have $m$ kids in the room. We give a cookie to everyone (if we can). Then we do it again, and again, doing a full round each time. The number of cookies left in the box is the remainder when $a$ is divided by $m$, it is what's left over.
For example, if $a=40$ and $m=12$, we do $3$ full rounds, each kid gets $3$ cookies. This $3$ is called the quotient. We will have $4$ cookies left over, the remainder is $4$, in symbols $40\operatorname{\%} 12=4$. If we start with $72$ cookies, the remainder is $0$.
But if we start with $5$ cookies, then we can't even get started, we cannot distribute cookies without causing a riot. So the quotient is $0$, nobody gets a cookie. And all the cookies are left over, the remainder is $5$, that is, $5\operatorname{\%}12=5$.
I think maybe you've not fully understood the modulo operator. Maybe this picture will help:
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 $120$.
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? $113\cdot 17 = 1921 = 120\cdot 16+1$