Modern algebra help!!

Write the set of solutions of
x≡ 5 (mod 24)
x≡ 17 (mod 18)

if any, as the solutions to a single congruence

x≡ 5 (mod 24)
<==> x≡ 5 (mod 3) and x≡ 5 (mod 8)
<==> x≡ 2 (mod 3) and x≡ 5 (mod 8)

x≡ 17 (mod 18)
<==> x≡ 17 (mod 9) and x≡ 17 (mod 2)
<==> x≡ 8 (mod 9) and x≡ 1 (mod 2)

Since x≡ 2 (mod 3) and x≡ 8 (mod 9) ==> x≡ 8 (mod 9),
and x≡ 5 (mod 8) and x≡ 1 (mod 2) ==> x≡ 5 (mod 8),

The above system with non-relatively prime moduli does have a solution,
having reduces it to x≡ 8 (mod 9) and x≡ 5 (mod 8).

With a little effort, we see that x ≡ 53 (mod 72) is the general solution.

I hope this helps!