Question

Question #6d78f

Answer 1

`2*4-=1" (mod "7")"`, thus `4=2^(-1)` in `ZZ_7`.

Explanation:

(This answer assumes a basic understanding of modular arithmetic)

When working in `ZZ_n`, if `k in ZZ_n` is coprime with `n`, that is, if `"GCD"(k, n) = 1` then there exists a unique `k^(-1) in ZZ_n` such that `k*k^(-1) -= 1" (mod "n")"`. We call `k^(-1)` the multiplicative inverse of `k` in `ZZ_n`.

To see that `2^(-1) -= 4` in `ZZ_7`, then, we need only show that `2*4 -= 1" (mod "7")"`. Indeed,

`2*4-=8-=1+7-=1" (mod "7")"`

Thus `2^(-1) = 4` in `ZZ_7`