How many 3-digit numbers have the property that the units digit is at least twice the tens digit?
I need to find all the 3-digit numbers that satisfy the condition that the units digit is at least twice the tens digit. I can think of some examples that work, like 132, 164, 182, and 200. I can also think of some examples that don't work, like 123, 143, 171, and 196. I notice that the hundreds digit doesn't seem to matter, as long as it is not zero. So I can focus on the last two digits and see what patterns emerge. I see that if the tens digit is 0, then the units digit can be anything from 0 to 9, giving me 10 choices. If the tens digit is 1, then the units digit can be anything from 2 to 9, giving me 8 choices. If the tens digit is 2, then the units digit can be anything from 4 to 9, giving me 6 choices. If the tens digit is 3, then the units digit can be anything from 6 to 9, giving me 4 choices. If the tens digit is 4, then the units digit can be anything from 8 to 9, giving me 2 choices. If the tens digit is 5 or more, then the units digit cannot be twice as large or larger, so there are no choices. Therefore, the total number of 3-digit numbers that have the desired property is 10 + 8 + 6 + 4 + 2 = 30 for each possible hundreds digit from 1 to 9. So the final answer is 30 times 9, which is 270. 