For what digit $d$ is the five-digit number $2345d$ a multiple of 9?
In order for a number to be a multiple of 9, the sum of its digits must be divisible by 9. Since $2+3+4+5=14$, the only single digit that will make the sum a multiple of 9 is $4$. The sum of the digits would be $18$, which is $9\cdot 2$, so $d=\boxed{4}$.
4
Level 2
Prealgebra
Problem: For what digit $d$ is the five-digit number $2345d$ a multiple of 9?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the digit $d$ that makes the number $2345d$ a multiple of 9, we need to apply the rule for divisibility by 9. This rule states that a number is divisible by 9 if the sum of its digits is divisible by 9. 

First, we calculate the sum of the known digits in the number $2345d$. The sum of the digits $2$, $3$, $4$, and $5$ is:

2 + 3 + 4 + 5 = 14.

Next, we include the digit $d$ in this sum, resulting in:

14 + d.

Now, we need to find a digit $d$ (where $d$ can be any integer from 0 to 9) such that the total sum $14 + d$ is divisible by 9. 

To find suitable values for $d$, we can check the possible sums:

- If $d = 0$, then $14 + 0 = 14$ (not divisible by 9).
- If $d = 1$, then $14 + 1 = 15$ (not divisible by 9).
- If $d = 2$, then $14 + 2 = 16$ (not divisible by 9).
- If $d = 3$, then $14 + 3 = 17$ (not divisible by 9).
- If $d = 4$, then $14 + 4 = 18$ (divisible by 9).
- If $d = 5$, then $14 + 5 = 19$ (not divisible by 9).
- If $d = 6$, then $14 + 6 = 20$ (not divisible by 9).
- If $d = 7$, then $14 + 7 = 21$ (not divisible by 9).
- If $d = 8$, then $14 + 8 = 22$ (not divisible by 9).
- If $d = 9$, then $14 + 9 = 23$ (not divisible by 9).

From this analysis, the only value of $d$ that results in a sum divisible by 9 is $d = 4$. 

Thus, the digit $d$ that makes the five-digit number $2345d$ a multiple of 9 is $4$.<|END_OF_LATENT|> The final answer is $4$.