Solving the Congruence $20x \equiv 16 \pmod{92}$ and Giving Answer As a Congruence to the Smallest Possible Modulus

Question

I have the following problem:

Solve the congruence $20x \equiv 16 \pmod{92}$. Give your answer as

(i) a congruence to the smallest possible modulus;

(ii) a congruence modulo $92$.

I just recently solved another congruence equations problem:

Solve the following congruences, or explain why they have no solution:

(i) $28x \equiv 3 \pmod{67}$;

(ii) $29x \equiv 3 \pmod{67}$.

I'm confused about how to solve (i) for the first problem. I usually solve these problems by using (1) the Euclidean algorithm to find the greatest common divisor, and (2) then using the extended Euclidean algorithm. But how would the way you solve (i) in the first problem differ from how you solve the second problem?

Also, would I be correct in saying that solving (ii) of the first problem is just done in the same way you solve the second problem?

I found this related question on congruences to the smallest possible modulus, but it doesn't seem to actually explain anything; it just goes straight to some calculations.

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

Yor first equation can by written as $$5x\equiv 4\mod 23$$, then you can write $$x\equiv \frac{4}{5}\equiv \frac{27}{5}\equiv \frac{50}{5}\equiv 10\mod 23$$