Find the numbers that have an inverse modulo 11

Question

I am trying to understand the inverse of a modulo.

I want to find the numbers in the range $1,2,3...11$ modulo $11$ that has an inverse.

I am confused and I can't understand how to identify which numbers modulo $11$ has an inverse.

Is the above exactly the same as the relatively prime numbers of 411$ ?

Answer 1

Yes, only numbers which are relatively prime to 11 will have an inverse $\mod {11}$. Of, course that would be all numbers $\{1,\ldots, 10\}$.

To find the inverse of a number $a\pmod{11}$ must find a number $n$ such that $an\equiv 1\pmod{11}$, or equivalently a pair of numbers such that $an+11m=1$. So use the Euclidean algorithm to find these numbers.

Answer 2

HINT:

Yes, to your last question. Now: Which numbers are relatively prime to $11$?

In general, the group of units modulo a prime $p$ contains all elements $\{1, 2, 3, \ldots, (p-1)\}$.

Answer 3

Yes, notice $\ \gcd(a,11) =1\overset{\rm Bezout}\iff \exists x,y\!:\ ax+11y = 1\iff \exists x\!:\ ax\equiv 1\pmod{11}$