The least number which leaves remainders 5, 9 and 17 when divided by 12, 16 and 24 respectively is:

Notice each remainder falls short of its divisor by the same amount: 12 − 5 = 16 − 9 = 24 − 17 = 7. So the number is 7 less than a common multiple. LCM(12, 16, 24) = 48, giving 48 − 7 = 41. Check: $41=12\times 3+5=16\times 2+9=24\times 1+17$. Adding 7 instead of subtracting gives the wrong answer 55.