A number leaves a remainder of 17 when divided by 22, and a remainder of 28 when divided by 33.
What is the smallest such positive number?

Here 22 - 17 = 5 and 33 - 28 = 5, so the number falls 5 SHORT of a common multiple. LCM(22, 33) = 66, so the number is 66 - 5 = 61. Check: $61=22\times 2+17$ and $61=33\times 1+28$. Adding 5 instead of subtracting gives 71; 66 is the LCM itself; 127 uses the next common multiple, 132.