Modular Arithmetic - Division

Question

How does one calculate (a/b) % m where a = x1*x2*.... and the numbers x1, x2,.. are quite large.

If we only had to find a % m , we could have easily done that using (x1%m) * (x2%m) *... but if there's something in the denominator 'b' in our case, how does one go about calculating it ?

I read somewhere this being done as (a % (m*b)) / b . I have been wondering if this is true and how do we go about proving it ?

Is *m* (known to be) a prime number by any chance? That makes the calculation simpler. — Pieter Geerkens, Sep 30 '14 at 21:53
Is b known to be a factor of a? Else, is it correct to assume integer division? — John Bollinger, Sep 30 '14 at 22:05
@PieterGeerkens. The value of m is not known. I was wondering if this was true for the general case. — kash, Oct 01 '14 at 04:51
but as @JohnBollinger suggested ! yes it seems the simple formula is only true if b is a factor of a. — kash, Oct 01 '14 at 04:52

score 0 · Accepted Answer · answered Sep 30 '14 at 22:56

You can get at least this far:

b * ((a / b) % m) = (b * (a / b)) % (b * m)
                  = (a - (a % b)) % (b * m)
                  = (a % (b * m) - (a % b) % (b * m))
                  = (a % (b * m) - (a % b))

hence,

((a / b) % m) = ((a % (b * m)) - (a % b)) / b

That is already easier to compute, hence I justify giving this as an answer. I think you can go on from there to the simpler formula you suggested, but I haven't the time or inclination to work that out. (So if this is homework then I leave a bit of it for you to do yourself :))

Modular Arithmetic - Division

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