Reverse of Modulo Operator

Question

I have posted this question originally in Stack Overflow.

The question is, "Is there a mathematical approach in getting the reverse of the Modulo Operator with given result $r$ and divisor $d$?"

So, the Modulo Operator % gives the remainder when dividing two numbers:

3 % 2 = 1

I will be asking the same question here.

Is there a mathematical approach in getting the reverse of the Modulo Operator with given result $r$ and divisor $d$?

NOTES: As I have stressed in the SO question, the answers should be limited to numbers which can be converted to a valid time in the format HHMMss.xxx where HH is the 24-hour respresention of hours, M is the minutes, s as seconds and x as millisecond. You need not worry about that part though because I can do a checking using Regex for that. You can check my updated question in SO.

if you write the definitions then the answer would be trivial. — jimjim, Mar 18 '18 at 14:22
Since $3\mathbin{%}2=1$ as well as $5\mathbin{%}2=1$, your question doesn't make sense. — egreg, Mar 18 '18 at 14:23
@egreg : that only shows that the reverse operation is not a function, the question does make sense. — jimjim, Mar 18 '18 at 14:27
Just write all numbers of the form $dk+r$ starting from $k=0$ until you're tired. What's the meaning of this? — egreg, Mar 18 '18 at 14:30
So in the integers the answer is the particular coset of the ideal generated by $d$. Pick your notation. $r+(d)$ or $r+d{Z}$. — Malcolm, Mar 18 '18 at 14:42
If you want the least $m$ such that $m \equiv r \pmod d$, then just set $m = r$. e.g., 1 % 2 = 1. — Robert Soupe, Mar 20 '18 at 15:05

F.A. · Accepted Answer · 2018-03-18T15:10:24.920

4

As pointed out in the comments and in the original Stack Overflow post, there are infinitely many numbers which divided by $\mathbf{d}$ have remainder $\mathbf{r}$. So there is a reverse operator given $d$ and $r$, but it will give you infinitely many solutions of the form $d\cdot k +r$.

edited Mar 18 '18 at 15:10

answered Mar 18 '18 at 14:33

F.A.

1,001

I would like to point out that the answers, as I know that there could be multiple answers, should be less than 240000000. that will limit the "infinite" answers, right? Moreover, I would like to convert the numbers to a time format, so, [00 - 23][00-59][00-59]. But of course, you need not think about the time formatting. I can do that using code. – xGeo Mar 20 '18 at 16:54
1

@Geoman I guess you cold write a $while$ loop and make one of the conditions be $d\cdot k + r < 240000000$ while you keep increasing $k$. – F.A. Mar 20 '18 at 17:47
And yes, then there would be a finite number of answers – F.A. Mar 20 '18 at 17:57
nice suggestion @F.A. I think that will be faster. Thanks! – xGeo Mar 20 '18 at 19:14

Reverse of Modulo Operator

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