Question

I know this may seem like a math question but i just saw this in a contest and I really want to know how to solve it.

We have

a (mod c)

and

b (mod c)

and we’re looking for the value of the quotient

(a/b) (mod c)

Any ideas?

Answer 1

In the ring of integers modulo C, these equations are equivalent:

A / B (mod C)
A * (1/B) (mod C)
A * B^-1(mod C).

Thus you need to find B^-1, the multiplicative inverse of B modulo C. You can find it using e.g. extended Euclidian algorithm.

Note that not every number has a multiplicative inverse for the given modulus.

Specifically, B^-1 exists if and only if gcd(B, C) = 1 (i.e. B and C are coprime).

See also

• Wikipedia/Modular multiplicative inverse
• Wikipedia/Extended Euclidian algorithm

---

Modular multiplicative inverse: Example

Suppose we want to find the multiplicative inverse of 3 modulo 11.

That is, we want to find

x = 3^-1(mod 11)
x = 1/3 (mod 11)
3x = 1 (mod 11)

Using extended Euclidian algorithm, you will find that:

x = 4 (mod 11)

Thus, the modular multiplicative inverse of 3 modulo 11 is 4. In other words:

A / 3 == A * 4 (mod 11)

---

Naive algorithm: brute force search

One way to solve this:

3x = 1 (mod 11)

Is to simply try x for all values 0..11, and see if the equation holds true. For small modulus, this algorithm may be acceptable, but extended Euclidian algorithm is much better asymptotically.