Determine the integers $a$ which have a multiplicative inverse:
(a) modulo 12
(b) modulo 13
(c) modulo 15
(a) $ax \equiv 1 \pmod {12}$ is the equation we want to solve.
We require $\gcd(12,a)=1$ for there to be a unique solution.
This limits the values of $a$ to 1, 5, 7, 11.
(b) $ax \equiv 1 \pmod {13}$ is the equation we want to solve.
We require $\gcd(13,a)=1$ for there to be a unique solution. Since 13 is prime, then any integer greater than 0 is co-prime to it,
The values of $a$ are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12.
(c) (a) $ax \equiv 1 \pmod {15}$ is the equation we want to solve.
We require $\gcd(15,a)=1$ for there to be a unique solution.
This limits the values of $a$ to 1, 2, 4, 7, 8, 11, 13, 14.