Question

What is `4^(-1)%15`?

Answer 1

4

Explanation:

The modular inverse `A^-1` of a number `A (mod b)` is the integer between 0 and `b-1` such that satisfies the equality

`A * A^-1 (mod b) = 1`

For this particular case, we have:

`4 * x (mod 15) = 1`

This means that:

`4x = 1 + 15n` for some integer `n`

If we let n = 1, we see that:

`4x = 1 + 15(1)`

`4x = 16`

`x=4`

So we can say that 4 is its own inverse, mod 15.

Final Answer

Answer 2

I assume that you mean `(4%15)^(-1)`, rather than `(4^(-1))%15`.

Well, normally, if we take `a % b`, then we get the remainder of dividing `a` by `b`. For instance, `4 mod 3 = 1`, because `3` divides once into `4` with a remainder of `1`.

But if `b > a`, we have that `a % b` returns `a`. Thus, this is the same as taking the inverse of `4`. Any number can be treated as a `1xx1` matrix.

We necessarily have that a `1xx1` inverse is just the reciprocal:

`[4] cdot [4]^(-1) = [1]`,

or that `color(blue)((4%15)^(-1)) = (4)^(-1) = color(blue)(1/4)`, as seen here.

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If in fact you mean the inverse of `4`, subsequently mod `15`, then we are saying

`(4^(-1))%15 = 4`

as seen here and in John D.'s answer.