Find the inverse of 35 in $$\mathbb{Z}/24\mathbb{Z}$$.

Question

anonymous · Answer

35 is equivalent to 11 mod 24.  so we need to find the inverse of 11.

We are looking for the solution to:
$$11x \equiv 1$$
mod 24.

You can do this with trial an error, or by computing the Bezout coefficients.

anonymous · Answer

!!!!!!!!

anonymous · Answer

but $$x \neq1/11$$ right..?

anonymous · Answer

hm...i think i'll do it the Bezout coefficient way.  The gcd of 24 and 11 is 1, so according to Bezout, there exist some integers x and  y such that:
$$24x+11y = 1$$

Those integers turn out to be x = -5, y = 11.  Sow now we have:

$$24(-5)+11(11) = 1$$

taking the whole equation mod 24 leaves us with:

$$11(11) \equiv 1$$
so the inverse of 11, is 11 itself.

anonymous · Answer

right, you cant just divide when talking about modular arithmetic.

anonymous · Answer

it takes a little more thinking and computing, to get inverses.  and some numbers dont even have inverses. (well, depending of what mod you are computing.)

anonymous · Answer

hmmm okay. i think i get it. thanks (:

anonymous · Answer

if you have any more questions just ask. :)

anonymous · Answer

The question should have specified whether you were to find the multiplicative or additive inverse.