If we have to solve for x in $3x ≡ 2 \pmod 5$, then we cannot just immediately divide both sides by 3. Because $\frac{2}3$ is not an integer.
Instead, we have to add multiples of 5 to either 3x or 2.
In this case, we can add 5 × 2 to 2.
$$3x ≡ 2 + 10 \pmod {5}$$ $$3x ≡ 12 \pmod {5}$$ Considering $gcd(3, 5) = 1$, we can divide both sides by 3.
$$x ≡ 4 \pmod {5}$$
But what happens if we cant easily find what number to add? In the first example, we easily know what to add, which is 10.
We have to solve for x in $83x ≡ 105 \pmod {120}$
Adding multiples of 120 to 105 is a tedious task to find if the result is a multiple of 83.
So my question is: How do we easily find what number to add?
