### Instruction: The positive three-digit integer $N$ has a ones digit of $0$. What is the probability that $N$ is divisible by $4$? Express your answer as a common fraction.### Response: A three-digit number with a ones digit of $0$ must be a multiple of $10$.
To be divisible by $4$, a multiple of $10$ must also be divisible by $2$ in its tens digit.
That means the tens digit of $N$ must be even.
There are five possible even digits: $0, 2, 4, 6, 8$.
The hundreds digit of $N$ can be any non-zero digit, since it does not affect the divisibility by $4$.
There are nine possible non-zero digits: $1, 2, 3, 4, 5, 6, 7, 8, 9$.
Therefore, the total number of three-digit numbers with a ones digit of $0$ is $9 \times 10 = 90$.
Of these, the number of numbers that are divisible by $4$ is $9 \times 5 = 45$.
The probability of selecting such a number at random is $\frac{45}{90} = \frac{1}{2}$.