What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits?
For a number to be divisible by 9, the sum of its digits must be divisible by 9.  But since the number has two even digits and two odd digits, the sum of its digits is even.  Thus the sum of its digits must be at least 18.  This number will be minimized if its thousands digit is 1 and its hundreds digit is 0.  This means the remaining two digits must sum to 17, and are hence 8,9.  So we see that the smallest possible integer of the desired form is $\boxed{1089}$.
1089
Level 5
Number Theory
Problem: What is the smallest positive four-digit number divisible by 9 which has two even and two odd digits?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the smallest positive four-digit number divisible by 9 with two even and two odd digits, we start by identifying the properties of four-digit numbers. The smallest four-digit number is 1000. 

Next, we need to ensure the number is divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9. 

We also need to consider the distribution of even and odd digits. The even digits available are 0, 2, 4, 6, and 8, while the odd digits are 1, 3, 5, 7, and 9. 

To satisfy the requirement of having two even and two odd digits, we can start constructing numbers from the smallest four-digit number, 1000, and incrementally check each number. 

The first four-digit number is 1000, which has the digits 1, 0, 0, and 0. This has three even digits and one odd digit, so it does not meet the criteria. 

Next, we check 1001, which has the digits 1, 0, 0, and 1. This also does not meet the criteria as it has two odd and two even digits but is not divisible by 9. 

Continuing this process, we check each subsequent number, ensuring to count the even and odd digits and calculate the sum of the digits to check for divisibility by 9. 

After checking several numbers, we reach 1089. The digits of 1089 are 1, 0, 8, and 9. Here, we have two odd digits (1 and 9) and two even digits (0 and 8). 

Next, we calculate the sum of the digits: 1 + 0 + 8 + 9 = 18. Since 18 is divisible by 9, 1089 meets the divisibility requirement. 

Thus, after verifying the conditions of having two even and two odd digits and confirming divisibility by 9, we conclude that the smallest positive four-digit number that satisfies all these conditions is 1089.<|END_OF_LATENT|> The final answer is $1089$.