A number gives remainders 1 and 3 when divided successively by 3 and 5. If the same number is divided successively by 5 and 3, the remainders are:

Rebuild the number: the first quotient is 5k + 3, so the number is 3(5k + 3) + 1 = 15k + 10. Take the smallest, 10. Dividing successively by 5 then 3: 10 ÷ 5 = 2 remainder 0, then 2 ÷ 3 = 0 remainder 2. So the remainders are 0 and 2 — and the same happens for 25, 40 and so on. Simply swapping the given pair to '3 and 1' is the trap.