discrete mathematics - Solve the congruence $9x equiv −3 pmod{24}$. Give your answer as a congruence to the smallest possible modulus, and as a congruence modulo 24.

I've been able to find the answer as a congruence to the smallest possible modulus (i.e. mod 8) but unsure how to find answer as congruence to mod 24. Also, is everything I've done below correct?:

gcd(9,24) = 3

Therefore, our congruence becomes 3x ≡ -1 (mod 8)

So, 3x ≡ 7 (mod 8)

We must find inverse 'c' of 3 (mod 8), i.e. 3c ≡ 1(mod 8)

gcd(3,8) = 1

let 3c + 8y = 1

Using extended Euclidean Algorithm, we get c = 1

Therefore, solution of 3x ≡ 7 (mod 8) (i.e. smallest possible modulus) is:

x ≡ 7 (mod 8)

Now, how to find solution as a congruence to modulus 24? Assuming everything I've done above is correct.

Answer

How do you get $c=1$. The inverse of $3c \equiv 1 \pmod{8}$ is $c=3$ (since $3\times 3=9$). In this way, you obtain $x \equiv 5 \pmod{8}$.

Observe that $9 \times 5=45$ and $24 \times 2=48$, so $x \equiv 5 \pmod{24}$.