A number gives remainders 2 and 3 when divided successively by 5 and 4. What remainder does it leave when divided by 20?

Work backwards through the two divisions. The second step says the first quotient is $4p+3$, and the first step says $N=5$ × (that quotient) + $2=5(4p+3)+2=20p+17$. So the remainder on division by 20 is 17. (The smallest such number is 17 itself: 17 ÷ 5 = 3 remainder 2, then 3 ÷ 4 = 0 remainder 3.) Doing the divisions in the other order would give 11 — a common slip.