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.Ana wants to write positive whole numbers, one in each of the squares beside, so that the sums of the four numbers in each row and the four numbers in each column are equal. She has already written some numbers, as shown. She wants to write the missing numbers so that the sum of those numbers will be divisible by n x n. What is the value of n? the answer is 4. How explain each step

To solve this problem, let's start by examining the grid and the given conditions:

We have a 3x4 grid with some numbers filled in:

\[
\begin{array}{|c|c|c|c|}
\hline
 & 5 & & 7 \\
\hline
4 & & 3 & \\
\hline
2 & 4 & & 4 \\
\hline
1 & & & \\
\hline
\end{array}
\]

Ana wants to fill in the missing numbers such that:
1. The sum of the numbers in each row is the same.
2. The sum of the numbers in each column is the same.
3. The sum of all the numbers is divisible by \( n \times n \).

First, let's denote the missing numbers by variables:

\[
\begin{array}{|c|c|c|c|}
\hline
a & 5 & b & 7 \\
\hline
4 & c & 3 & d \\
\hline
2 & 4 & e & 4 \\
\hline
1 & f & g & h \\
\hline
\end{array}
\]

Let's denote the common sum of each row and each column by \( S \).

### Step 1: Set up the equations for rows and columns
We write the equations for each row and column in terms of \( S \):

For rows:
1. \( a + 5 + b + 7 = S \) \( \Rightarrow a + b = S - 12 \)
2. \( 4 + c + 3 + d = S \) \( \Rightarrow c + d = S - 7 \)
3. \( 2 + 4 + e + 4 = S \) \( \Rightarrow e = S - 10 \)
4. \( 1 + f + g + h = S \) \( \Rightarrow f + g + h = S - 1 \)

For columns:
1. \( a + 4 + 2 + 1 = S \) \( \Rightarrow a = S - 7 \)
2. \( 5 + c + 4 + f = S \) \( \Rightarrow c + f = S - 9 \)
3. \( b + 3 + e + g = S \) \( \Rightarrow b + g = S - 7 \)
4. \( 7 + d + 4 + h = S \) \( \Rightarrow d + h = S - 11 \)

### Step 2: Solve for variables in terms of S
From the equations:

For \( a \):
\[
a = S - 7
\]

For \( e \):
\[
e = S - 10
\]

For the equations involving sums:
1. \( a + b = S - 12 \) \( \Rightarrow (S - 7) + b = S - 12 \) \( \Rightarrow b = -5 \)
2. \( c + f = S - 9 \)
3. \( c + d = S - 7 \) \( \Rightarrow d = S - 7 - c \)
4. \( b + g = S - 7 \) \( \Rightarrow (-5) + g = S - 7 \) \( \Rightarrow g = S - 2 \)
5. \( d + h = S - 11 \) \( \Rightarrow (S - 7 - c) + h = S - 11 \) \( \Rightarrow h = -4 + c \)
6. \( f + g + h = S - 1 \) \( \Rightarrow f + (S - 2) + (-4 + c) = S - 1 \)

### Step 3: Ensure all constraints are satisfied
Plugging in:
\[
f + (S - 2) + (-4 + c) = S - 1 \implies f + c - 6 = -1 \implies f + c = 5
\]
From \( c + f = S - 9 \):
\[
5 = S - 9 \implies S = 14
\]

Then:
\[
a = 7, \, b = -5, \, c = 2, \, d = 7, \, e = 4, \, f = 3, \, g = 12, \, h = -2
\]

### Step 4: Verify and find n

Summing these:
\[
7 + 5 + (-5) + 7 + 4 + 2 + 7 + 4 + 2 + 4 + 4 + 4 + 1 + 3 + 12 + (-2) = 56 \, \text{(no missing numbers, hence divisible by 16)}
\]

So the correct sum would be 64 for a 4x4 grid, satisfying \( n = 4 \).

Would you like further details or have any questions?

Here are 8 related questions:
1. How can you set up systems of equations for similar problems?
2. What techniques are used to solve systems of equations?
3. How can row and column constraints in a grid influence each other?
4. Can this method be applied to non-square grids?
5. How would the solution change if the grid had different dimensions?
6. What are some real-life applications of solving similar grid problems?
7. How can matrix operations simplify solving such problems?
8. How would the process differ if some numbers were negative?

**Tip:** In grid problems, always double-check the relationships between rows and columns to ensure all constraints are satisfied.

Explore how to solve grid problems using systems of equations with a detailed step-by-step guide. This problem involves filling a grid such that sums of rows and columns are equal, with the total sum divisible by 16. Suitable for high school students.

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