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QUESTION: I am experimenting with abstract math (for fun, not for school), and I am curious about the modulo operator (%) that is commonly used in programming. Is there some kind of more elementary function that makes it up, or is it a purely arbitrary construction? I guess another way to express it is this: If I want to graph (for example) y = x % 3 but my calculator doesn’t have a modulus operator, is there some way I can get the same result using some combination of the other available function operators? (+ - × ÷ log exp sin etc.)

ANSWER: The modulus operation gives the remainder. Your example of y = x % 3 means that y is the remainder when you divide x by 3.

So 14 mod 6 is 2, because 14/2 is 2 with remainder 2., i.e. 14 = 2*6 + 2.

There are two ways to do this using the standard functions.

First, you can divide and then multiply:

You would type 14/6 to get 2.33333.

Then subtract away the integer part to get 0.333333.

Then multiply back by 6 to get 2. That's the remainder.

The other method is subtracting 6 repeatedly: 14 - 6 = 8. 8 - 6 = 2. You stop when you get a number less than 6 (so it is 2 again).

---------- FOLLOW-UP ----------

QUESTION: Thank you. I get the part about doing it with specific numbers such as 14 and 6, but is there a general way to do it with variables such as x and y? In the case of 14 mod 6, it’s visibly obvious how much of an integer to subtract (2) before multiplying by 6 to obtain a remainder of 2. But in the case of y = x mod c, how do you know how much of an integer to subtract from x ÷ c before multiplying by c to obtain the remainder?

If you want to compute y = x % m, you are trying to divide x by m.

You can do x/m, then subtract away the whole-number-part (like 2.3333 becomes 0.3333), then multiply by m again.

You can subtract m from x repeatedly until you get a number between 0 and m-1.

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