Multiplication table from 1 to 12, ideal for math practice and education.

Problem Description:

The image shows a multiplication table with rows and columns labeled from 1 to 12. The task is to fill in the missing values in the table by performing the multiplications.

Solution Approach:

To solve this problem, we need to compute the product of each row label and each column label for every cell in the table. The general formula for any cell at the intersection of row \( r \) and column \( c \) is:
\[
\text{Value} = r \times c
\]
where \( r \) is the row number and \( c \) is the column number.

Step-by-Step Solution:

We will compute the values for each section of the table systematically.

#### First Section (Rows 1 to 4):
- Row 1: \( 1 \times 1 = 1, 1 \times 2 = 2, \ldots, 1 \times 12 = 12 \)
- Row 2: \( 2 \times 1 = 2, 2 \times 2 = 4, \ldots, 2 \times 12 = 24 \)
- Row 3: \( 3 \times 1 = 3, 3 \times 2 = 6, \ldots, 3 \times 12 = 36 \)
- Row 4: \( 4 \times 1 = 4, 4 \times 2 = 8, \ldots, 4 \times 12 = 48 \)

#### Second Section (Rows 5 to 8):
- Row 5: \( 5 \times 1 = 5, 5 \times 2 = 10, \ldots, 5 \times 12 = 60 \)
- Row 6: \( 6 \times 1 = 6, 6 \times 2 = 12, \ldots, 6 \times 12 = 72 \)
- Row 7: \( 7 \times 1 = 7, 7 \times 2 = 14, \ldots, 7 \times 12 = 84 \)
- Row 8: \( 8 \times 1 = 8, 8 \times 2 = 16, \ldots, 8 \times 12 = 96 \)

#### Third Section (Rows 9 to 12):
- Row 9: \( 9 \times 1 = 9, 9 \times 2 = 18, \ldots, 9 \times 12 = 108 \)
- Row 10: \( 10 \times 1 = 10, 10 \times 2 = 20, \ldots, 10 \times 12 = 120 \)
- Row 11: \( 11 \times 1 = 11, 11 \times 2 = 22, \ldots, 11 \times 12 = 132 \)
- Row 12: \( 12 \times 1 = 12, 12 \times 2 = 24, \ldots, 12 \times 12 = 144 \)

Filled Multiplication Table:

Here is the completed multiplication table:

| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|----|----|----|----|----|----|----|----|----|----|----|----|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 10| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100| 110| 120|
| 11| 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110| 121| 132|
| 12| 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108| 120| 132| 144|

Final Answer:

\[
\boxed{
\begin{array}{c|cccccccccccc}
& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
2 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & 22 & 24 \\
3 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 27 & 30 & 33 & 36 \\
4 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 & 40 & 44 & 48 \\
5 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 & 55 & 60 \\
6 & 6 & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54 & 60 & 66 & 72 \\
7 & 7 & 14 & 21 & 28 & 35 & 42 & 49 & 56 & 63 & 70 & 77 & 84 \\
8 & 8 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72 & 80 & 88 & 96 \\
9 & 9 & 18 & 27 & 36 & 45 & 54 & 63 & 72 & 81 & 90 & 99 & 108 \\
10 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 & 110 & 120 \\
11 & 11 & 22 & 33 & 44 & 55 & 66 & 77 & 88 & 99 & 110 & 121 & 132 \\
12 & 12 & 24 & 36 & 48 & 60 & 72 & 84 & 96 & 108 & 120 & 132 & 144 \\
\end{array}
}
\]