To solve the system of congruences:

1. $20x \equiv 24 \mod 164$
2. $x \equiv 15 \mod 73$

Step 1: Solve the first congruence $20x \equiv 24 \mod 164$

We start by simplifying this congruence. Divide both sides by the greatest common divisor (GCD) of 20 and 164, which is 4:

$\frac{20x}{4} \equiv \frac{24}{4} \mod \frac{164}{4}$

This simplifies to:

$5x \equiv 6 \mod 41$

Next, we need to solve this linear congruence. We find the multiplicative inverse of 5 modulo 41. The inverse $y$ satisfies:

$5y \equiv 1 \mod 41$

We can find $y$ using the extended Euclidean algorithm, which gives:

$y = 33 \quad \text{since} \quad 5 \times 33 = 165 \equiv 1 \mod 41$

Now, multiply both sides of ( 5x