complete the scrambled multiplication table.

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Accepted Answer

To solve the scrambled multiplication table, we use the basic multiplication rule $ a\times b = c $, where $ a $ is the row number, $ b $ is the column number, and $ c $ is the product. We'll fill in the missing values row by row or column by column, using known products to find unknown factors.

### Step 1: Identify Row and Column Headers
First, we determine the row and column headers (the first column and first row, excluding the "×" cell) by finding the factors of the known products. For example:
- In the row with "4" (first column), the product with column "5" is $ 4\times5 = 20 $ (matches), with column "6" is $ 4\times6 = 24 $ (matches), and with column "3" is $ 4\times3 = 12 $ (matches). So row "4" is correct.
- In the column with "5" (first row), the product with row "5" (let's check the row with product "25" in column "5": $ 5\times5 = 25 $), so row "5" (first column) has "5".
- In the column with "2" (first row), the product with row "8" is $ 8\times2 = 16 $ (matches), with row "5" is $ 5\times2 = 10 $ (matches), so column "2" is correct.

### Step 2: Fill in Missing Column Headers (First Row)
The first row (column headers) has missing values. Let's find them using row products:
- For the cell in row "4", column (unknown): $ 4\times x = 36 $ ⇒ $ x = \frac{36}{4} = 9 $. So column "9" (first row) is 9.
- For the cell in row "1", column (unknown) with product "8": $ 1\times x = 8 $ ⇒ $ x = 8 $. So column "8" (first row) is 8.
- For the cell in row "1", column (unknown) with product "7": $ 1\times x = 7 $ ⇒ $ x = 7 $. So column "7" (first row) is 7.
- For the cell in row "1", column (unknown) with product "1": $ 1\times x = 1 $ ⇒ $ x = 1 $. So column "1" (first row) is 1.
- For the cell in row "9", column (unknown) with product "3": $ 9\times x = 3 $ ⇒ $ x = \frac{3}{9} = \frac{1}{3} $? No, wait—row "9" (first column) is actually "3"? Wait, no, row "9" (first column) has product "15" with column "5": $ 3\times5 = 15 $, so row "9" (first column) is "3" (since $ 3\times5 = 15 $). Let's correct: row "3" (first column) has "3", because $ 3\times5 = 15 $, $ 3\times2 = 6 $, $ 3\times1 = 3 $. So row "3" (first column) is 3.
- For the cell in row "7", column (unknown) with product "63": $ 7\times x = 63 $ ⇒ $ x = 9 $ (already known, column "9").
- For the cell in row "5", column (unknown) with product "5": $ 5\times x = 5 $ ⇒ $ x = 1 $ (column "1", already known).
- For the cell in row "6", column (unknown) with product "48": $ 6\times x = 48 $ ⇒ $ x = 8 $ (column "8", already known).
- For the cell in row "9" (first column "9"), column (unknown) with product "72": $ 9\times x = 72 $ ⇒ $ x = 8 $ (column "8", already known).
- For the cell in row "2" (first column "2"), column (unknown) with product "10": $ 2\times x = 10 $ ⇒ $ x = 5 $ (column "5", already known), product "8": $ 2\times x = 8 $ ⇒ $ x = 4 $ (column "4", already known), product "14": $ 2\times x = 14 $ ⇒ $ x = 7 $ (column "7", already known), product "6": $ 2\times x = 6 $ ⇒ $ x = 3 $ (column "3", already known), product "2": $ 2\times x = 2 $ ⇒ $ x = 1 $ (column "1", already known), product "12": $ 2\times x = 12 $ ⇒ $ x = 6 $ (column "6", already known). So row "2" (first column) is 2.

### Step 3: Fill in Missing Row Headers (First Column)
The first column (row headers) has missing values. Let's confirm:
- Row with product "25" in column "5": $ 5\times5 = 25 $, so row "5" (first column) is 5.
- Row with product "30" in column "5": $ 6\times5 = 30 $, so row "6" (first column) is 6.
- Row with product "45" in column "5": $ 9\times5 = 45 $, so row "9" (first column) is 9.
- Row with product "10" in column "5": $ 2\times5 = 10 $, so row "2" (first column) is 2.
- Row with product "5" in column "5": $ 1\times5 = 5 $, but row "1" (first column) has product "5" with column "5"? Wait, row "1" (first column) has product "5" with column "5": $ 1\times5 = 5 $, which matches.

### Step 4: Fill in Remaining Cells
Now, use $ \text{row} \times \text{column} $ for each empty cell:
- **Row 4, Column 8**: $ 4\times8 = 32 $
- **Row 4, Column 7**: $ 4\times7 = 28 $
- **Row 4, Column 1**: $ 4\times1 = 4 $
- **Row 4, Column 2**: $ 4\times2 = 8 $
- **Row 1, Column 5**: $ 1\times5 = 5 $
- **Row 1, Column 4**: $ 1\times4 = 4 $
- **Row 1, Column 3**: $ 1\times3 = 3 $
- **Row 1, Column 6**: $ 1\times6 = 6 $
- **Row 1, Column 2**: $ 1\times2 = 2 $
- **Row 8, Column 5**: $ 8\times5 = 40 $
- **Row 8, Column 9**: $ 8\times9 = 72 $
- **Row 8, Column 1**: $ 8\times1 = 8 $
- **Row 8, Column 3**: $ 8\times3 = 24 $ (already filled)
- **Row 8, Column 7**: $ 8\times7 = 56 $ (already filled)
- **Row 8, Column 6**: $ 8\times6 = 48 $
- **Row 3, Column 8**: $ 3\times8 = 24 $
- **Row 3, Column 4**: $ 3\times4 = 12 $
- **Row 3, Column 7**: $ 3\times7 = 21 $ (already filled)
- **Row 3, Column 6**: $ 3\times6 = 18 $
- **Row 3, Column 2**: $ 3\times2 = 6 $ (already filled)
- **Row 3, Column 1**: $ 3\times1 = 3 $ (already filled)
- **Row 7, Column 8**: $ 7\times8 = 56 $
- **Row 7, Column 4**: $ 7\times4 = 28 $ (already filled)
- **Row 7, Column 7**: $ 7\times7 = 49 $
- **Row 7, Column 3**: $ 7\times3 = 21 $ (already filled)
- **Row 7, Column 1**: $ 7\times1 = 7 $ (already filled)
- **Row 7, Column 2**: $ 7\times2 = 14 $ (already filled)
- **Row 5, Column 8**: $ 5\times8 = 40 $
- **Row 5, Column 9**: $ 5\times9 = 45 $
- **Row 5, Column 7**: $ 5\times7 = 35 $ (already filled)
- **Row 5, Column 3**: $ 5\times3 = 15 $
- **Row 5, Column 1**: $ 5\times1 = 5 $ (already filled)
- **Row 5, Column 2**: $ 5\times2 = 10 $ (already filled)
- **Row 5, Column 6**: $ 5\times6 = 30 $
- **Row 6, Column 4**: $ 6\times4 = 24 $
- **Row 6, Column 7**: $ 6\times7 = 42 $
- **Row 6, Column 2**: $ 6\times2 = 12 $
- **Row 6, Column 1**: $ 6\times1 = 6 $ (already filled)
- **Row 6, Column 3**: $ 6\times3 = 18 $ (already filled)
- **Row 6, Column 9**: $ 6\times9 = 54 $ (already filled)
- **Row 9, Column 5**: $ 9\times5 = 45 $
- **Row 9, Column 7**: $ 9\times7 = 63 $
- **Row 9, Column 3**: $ 9\times3 = 27 $ (already filled)
- **Row 9, Column 1**: $ 9\times1 = 9 $ (already filled)
- **Row 9, Column 6**: $ 9\times6 = 54 $
- **Row 2, Column 4**: $ 2\times4 = 8 $ (already filled)
- **Row 2, Column 9**: $ 2\times9 = 18 $
- **Row 2, Column 3**: $ 2\times3 = 6 $ (already filled)
- **Row 2, Column 1**: $ 2\times1 = 2 $ (already filled)
- **Row 2, Column 6**: $ 2\times6 = 12 $ (already filled)

### Final Table (Key Cells Filled)
| ×  | 5  | 8  | 4  | 9  | 7  | 3  | 1  | 2  | 6  |
|----|----|----|----|----|----|----|----|----|----|
| 4  | 20 | 32 | 16 | 36 | 28 | 12 | 4  | 8  | 24 |
| 1  | 5  | 8  | 4  | 9  | 7  | 3  | 1  | 2  | 6  |
| 8  | 40 | 64 | 32 | 72 | 56 | 24 | 8  | 16 | 48 |
| 3  | 15 | 24 | 12 | 27 | 21 | 9  | 3  | 6  | 18 |
| 7  | 35 | 56 | 28 | 63 | 49 | 21 | 7  | 14 | 42 |
| 5  | 25 | 40 | 20 | 45 | 35 | 15 | 5  | 10 | 30 |
| 6  | 30 | 48 | 24 | 54 | 42 | 18 | 6  | 12 | 36 |
| 9  | 45 | 72 | 36 | 81 | 63 | 27 | 9  | 18 | 54 |
| 2  | 10 | 16 | 8  | 18 | 14 | 6  | 2  | 4  | 12 |

(Note: This is a partial fill; the full table requires verifying all cells with $ \text{row} \times \text{column} $.)

# Answer:
The completed multiplication table (key cells) is as shown above, with each cell calculated as $ \text{row} \times \text{column} $.

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Question

Explanation:

Step 1: Identify Row and Column Headers

Step 2: Fill in Missing Column Headers (First Row)

Step 3: Fill in Missing Row Headers (First Column)

Answer: