Each of 18 cards is numbered with either a 4 or a 5. It turns out that the sum of all the numbers is divisible by 17. How many cards are labeled with a 4?

Sure, let's solve this step by step:

1. Let's denote the number of cards labeled with a 4 as x and the number of cards labeled with a 5 as y. We know that x + y = 18 because there are 18 cards in total.

2. We also know that the sum of all the numbers on the cards is divisible by 17. This gives us another equation: 4x + 5y is divisible by 17.

3. Now we have a system of two equations, and we can solve it. The easiest way is to use the method of substitution or the method of elimination.

4. If we rearrange the first equation, we get y = 18 - x. We can substitute this into the second equation: 4x + 5(18 - x) = 90 - x is divisible by 17.

5. Now we can try different values of x and see which one makes 90 - x divisible by 17. If we try x = 7, we get 90 - 7 = 83, which is not divisible by 17. If we try x = 8, we get 90 - 8 = 82, which is divisible by 17.

6. So, the number of cards labeled with a 4 is 8.