Key Concepts and Formulas

• The number of $n$-digit numbers is $9 \cdot 10^{n-1}$. This is because the first digit can be any of the digits from 1 to 9 (9 choices), and the remaining $n-1$ digits can be any of the digits from 0 to 9 (10 choices each).
• For a large enough set of numbers, roughly half will have an even digit sum and half will have an odd digit sum.
• Basic algebraic manipulation.

Step-by-Step Solution

Step 1: Find the total number of seven-digit numbers.

The smallest seven-digit number is $10^6 = 1,000,000$ and the largest is $10^7 - 1 = 9,999,999$. The total number of seven-digit numbers is $9,999,999 - 1,000,000 + 1 = 9,000,000 = 9 \cdot 10^6$. We can arrive at the same result by noting that the first digit can be any of the digits from 1 to 9 (9 choices), and the remaining 6 digits can be any of the digits from 0 to 9 (10 choices each). Therefore, the total number of seven-digit numbers is $9 \cdot 10^6$.

Step 2: Determine the number of seven-digit numbers with an even digit sum.

Since roughly half of all numbers have an even digit sum and half have an odd digit sum, the number of seven-digit numbers with an even digit sum is approximately half of the total number of seven-digit numbers. So, the number of seven-digit numbers with an even digit sum is approximately $\frac{1}{2} \cdot 9 \cdot 10^6 = 4.5 \cdot 10^6$.

Step 3: Rewrite the result in the required form.

We need to write $4.5 \cdot 10^6$ in the form $m \cdot n \cdot 10^n$, where $m, n \in \{1, 2, 3, \ldots, 9\}$. We can rewrite $4.5 \cdot 10^6$ as $45 \cdot 10^5$ or $9 \cdot 5 \cdot 10^5$. Thus, $m=9$ and $n=5$.

Step 4: Calculate $m+n$.

We have $m = 9$ and $n = 5$, so $m + n = 9 + 5 = 14$.

However, the question states that the answer is 7. There must be an error in the problem or the given answer.

Let's re-examine Step 3. We are given that the number of seven-digit numbers with an even digit sum is $m \cdot n \cdot 10^n$, where $m, n \in \{1, 2, 3, \ldots, 9\}$. We found that the number of such numbers is $4.5 \cdot 10^6 = 9 \cdot 5 \cdot 10^5$. Thus, $m=9$, $n=5$. But the problem statement says that $m+n=7$. The only way to make this work is if the number of seven-digit numbers with an even digit sum is actually $2 \cdot 5 \cdot 10^5 = 10^6$ instead of $4.5 \cdot 10^6$.

Since the answer is given to be 7, let's assume there is an error in the initial calculation, and work backwards. Suppose $m+n=7$. Then we can try some combinations of $m$ and $n$. If $m=1, n=6$, then $m \cdot n \cdot 10^n = 1 \cdot 6 \cdot 10^6 = 6 \cdot 10^6$. If $m=2, n=5$, then $m \cdot n \cdot 10^n = 2 \cdot 5 \cdot 10^5 = 10^6$. If $m=5, n=2$, then $m \cdot n \cdot 10^n = 5 \cdot 2 \cdot 10^2 = 10^3 = 1000$. If $m=6, n=1$, then $m \cdot n \cdot 10^n = 6 \cdot 1 \cdot 10^1 = 60$.

None of these are close to $4.5 \cdot 10^6$. We are given that the correct answer is $m+n=7$. We also know that the number of 7-digit numbers with even digit sum is $4.5 \cdot 10^6 = 9 \cdot 5 \cdot 10^5$.

Therefore, $m=9$ and $n=5$. So we have $m \cdot n \cdot 10^n = 9 \cdot 5 \cdot 10^5 = 4.5 \cdot 10^6$. Then $m+n = 9+5 = 14$.

The problem statement says that $m+n=7$. Let's re-examine the given form of the answer, $m \cdot n \cdot 10^n$. If we have $m=2$ and $n=5$, then $m \cdot n \cdot 10^n = 2 \cdot 5 \cdot 10^5 = 10^6$. This is wrong.

We have $4.5 \times 10^6 = \frac{9}{2} \times 10^6$. We are given that $m+n=7$. Suppose $n=6$. Then $m=1$. Then $m \cdot n \cdot 10^n = 1 \cdot 6 \cdot 10^6 = 6 \cdot 10^6$. Suppose $n=5$. Then $m=2$. Then $m \cdot n \cdot 10^n = 2 \cdot 5 \cdot 10^5 = 10^6$.

The only way to obtain $m+n=7$ is to assume that the number of 7-digit numbers with even digit sum is $10^6$. However, that's wrong.

There is likely an error in the problem statement. Assuming our calculation of the number of seven-digit numbers with an even digit sum is correct ($4.5 \cdot 10^6 = 9 \cdot 5 \cdot 10^5$), and assuming we have to write it in the form $m \cdot n \cdot 10^n$, where $m, n \in \{1, 2, 3, \ldots, 9\}$, then $m=9$ and $n=5$, so $m+n = 14$.

However, we are told that the correct answer is 7. So, we have to work backwards.

If $m+n=7$, then $m \cdot n \cdot 10^n$ can take the following values: $1 \cdot 6 \cdot 10^6 = 6 \cdot 10^6$ $2 \cdot 5 \cdot 10^5 = 10^6$ $3 \cdot 4 \cdot 10^4 = 12 \cdot 10^4 = 1.2 \cdot 10^5$ $4 \cdot 3 \cdot 10^3 = 12 \cdot 10^3 = 1.2 \cdot 10^4$ $5 \cdot 2 \cdot 10^2 = 10^3$ $6 \cdot 1 \cdot 10^1 = 60$

None of these are equal to $4.5 \cdot 10^6$.

Given the constraint that $m, n \in \{1, 2, 3, \ldots, 9\}$, and that the answer is 7, we can only conclude that there is an error in the problem statement. Despite our best efforts, we cannot reach the answer of 7 given the constraints.

Common Mistakes & Tips

• Always double-check the problem statement for any hidden constraints or conditions.
• When dealing with large numbers, it's often helpful to use scientific notation to simplify calculations.
• In problems involving digit sums, consider the properties of even and odd numbers.

Summary

The total number of seven-digit numbers is $9 \cdot 10^6$. The number of seven-digit numbers with an even digit sum is $\frac{1}{2} \cdot 9 \cdot 10^6 = 4.5 \cdot 10^6$. We can write this as $9 \cdot 5 \cdot 10^5$, so $m=9$ and $n=5$, and $m+n=14$. However, the problem states that $m+n=7$. There is likely an error in the problem statement or the provided answer. Given the constraints of the problem, we cannot arrive at the given answer of 7. The correct answer should be 14.

Final Answer

The final answer is \boxed{14}. Since none of the options are 14, there is an error in the problem statement or options.