Question

complete the scrambled multiplication table. table with columns/rows containing numbers like 8, 9, 7, 1, 32, 16, 28, 4, 8, 8, 4, 3, 2, 6, 40, 72, 8, 48, 3, 24, 12, 27, 9, 18, 56, 49, 7, 5, 40, 45, 15, 30, 24, 42, 12, 45, 63, 54, 2, 16, 18, 4 (table structure with missing cells to fill)

Explanation:

Step1: Identify the multiplier (row) and multiplicand (column)

In a multiplication table, each cell is the product of its row (let's call it \( r \)) and column (let's call it \( c \)), so \( \text{cell} = r \times c \). We can find \( r \) or \( c \) by dividing the known product by the known \( r \) or \( c \).

Step2: Analyze the row with 1 (column 6, value 4)

The row where the column with 1 has a value of 4: so \( r \times 1 = 4 \), so \( r = 4 \). Let's verify: column 2 (8) should be \( 4 \times 8 = 32 \) (matches), column 3 (16) is \( 4 \times 4 = 16 \) (so column 3 is 4), column 5 (28) is \( 4 \times 7 = 28 \) (so column 5 is 7), column 7 (4) is \( 4 \times 1 = 4 \) (matches), column 8 (8) is \( 4 \times 2 = 8 \) (so column 8 is 2).

Step3: Analyze the row with 3 (row 5)

Row 3: column 2 (8) gives \( 3 \times 8 = 24 \) (matches), column 3 (4) gives \( 3 \times 4 = 12 \) (matches), column 4 (9) gives \( 3 \times 9 = 27 \) (matches), column 6 (3) gives \( 3 \times 3 = 9 \) (so column 6 is 3), column 9 (6) gives \( 3 \times 6 = 18 \) (matches), column 8 (2) gives \( 3 \times 2 = 6 \) (so cell is 6).

Step4: Analyze the row with 5 (row 7)

Row 5: column 2 (8) gives \( 5 \times 8 = 40 \) (matches), column 4 (9) gives \( 5 \times 9 = 45 \) (matches), column 6 (3) gives \( 5 \times 3 = 15 \) (matches), column 9 (6) gives \( 5 \times 6 = 30 \) (matches), column 3 (4) gives \( 5 \times 4 = 20 \) (so cell is 20), column 5 (7) gives \( 5 \times 7 = 35 \) (so cell is 35), column 1 (let's find column 1: row 3, column 1: \( 3 \times c_1 =? \), row 4 (with 40 and 72): \( r \times 5 = 40 \) ⇒ \( r = 8 \); \( 8 \times 9 = 72 \) (matches). So row 4 is 8. Then row 4, column 1: \( 8 \times c_1 = 40 \) ⇒ \( c_1 = 5 \). So row 5, column 1: \( 5 \times 5 = 25 \).

Step5: Analyze row 2 (last row)

Row 2: column 2 (8) gives \( 2 \times 8 = 16 \) (matches), column 4 (9) gives \( 2 \times 9 = 18 \) (matches), column 8 (2) gives \( 2 \times 2 = 4 \) (matches), column 1 (5) gives \( 2 \times 5 = 10 \), column 3 (4) gives \( 2 \times 4 = 8 \), column 5 (7) gives \( 2 \times 7 = 14 \), column 6 (3) gives \( 2 \times 3 = 6 \), column 7 (1) gives \( 2 \times 1 = 2 \), column 9 (6) gives \( 2 \times 6 = 12 \).

Step6: Analyze row 8 (row 4)

Row 8: column 1 (5) gives \( 8 \times 5 = 40 \) (matches), column 4 (9) gives \( 8 \times 9 = 72 \) (matches), column 7 (1) gives \( 8 \times 1 = 8 \) (matches), column 9 (6) gives \( 8 \times 6 = 48 \) (matches), column 2 (8) gives \( 8 \times 8 = 64 \), column 3 (4) gives \( 8 \times 4 = 32 \), column 5 (7) gives \( 8 \times 7 = 56 \), column 6 (3) gives \( 8 \times 3 = 24 \).

Step7: Analyze row 7 (wait, row 5 is 5, row 6: let's find row with 56 and 49. \( r \times 8 = 56 \) ⇒ \( r = 7 \); \( 7 \times 7 = 49 \) (matches). So row 6 is 7. Then row 6, column 1 (5) gives \( 7 \times 5 = 35 \), column 3 (4) gives \( 7 \times 4 = 28 \), column 4 (9) gives \( 7 \times 9 = 63 \), column 6 (3) gives \( 7 \times 3 = 21 \), column 7 (1) gives \( 7 \times 1 = 7 \) (matches), column 8 (2) gives \( 7 \times 2 = 14 \), column 9 (6) gives \( 7 \times 6 = 42 \).

Step8: Analyze row with 8 (row 3? No, row 4 is 8, row with 5 (row 7), row with 3 (row 5), row with 4 (row 2? No, row 2 is 2). Wait, row with 8 (column 2, value 8) and 4 (column 3, value 4): \( r \times 8 = 8 \) ⇒ \( r = 1 \)? No, row with 8 (column 8, value 2) and 4 (column 3, value 4): \( r \times 4 = 4 \) ⇒ \( r = 1 \)? Wait, maybe better to list columns:

Columns (top to bottom, first row is column headers? Wait, first row: \( \times \), then 8, (empty), 9, 7, (empty), 1, (empty), (empty). Second row: (empty), 32, 16, (empty), 28, (empty), 4, 8, (empty). So column 2: 8 (header), 32 (row 2), 24 (row 5), 56 (row 6), 40 (row 7), (empty) (row 8), (empty) (row 9), 16 (row 10). So column 2 values: 8 (header), 32 (row2: 4×8), 24 (row5: 3×8), 64 (row4: 8×8), 56 (row6: 7×8), 40 (row7: 5×8), (empty) (row8: let's see row8 has 45 in column1: \( r \times 5 = 45 \) ⇒ \( r = 9 \). So row9 is 9. Then row9, column2: \( 9 \times 8 = 72 \), column3: \( 9 \times 4 = 36 \), column4: \( 9 \times 9 = 81 \), column5: \( 9 \times 7 = 63 \) (matches), column6: \( 9 \times 3 = 27 \), column7: \( 9 \times 1 = 9 \), column8: \( 9 \times 2 = 18 \), column9: \( 9 \times 6 = 54 \) (matches).

Step9: Row with 24 (column3, value24) and 42 (column5, value42): \( r \times 4 = 24 \) ⇒ \( r = 6 \); \( 6 \times 7 = 42 \) (matches). So row8 is 6. Then row8, column1: \( 6 \times 5 = 30 \), column2: \( 6 \times 8 = 48 \), column4: \( 6 \times 9 = 54 \), column6: \( 6 \times 3 = 18 \), column7: \( 6 \times 1 = 6 \), column8: \( 6 \times 2 = 12 \) (matches), column9: \( 6 \times 6 = 36 \).

Now, let's summarize the table (rows are \( r \), columns are \( c \) where \( c_1=5, c_2=8, c_3=4, c_4=9, c_5=7, c_6=3, c_7=1, c_8=2, c_9=6 \)):

- Row \( r=1 \): Wait, do we have \( r=1 \)? Let's check column7 (1): row2 (4) has 4, row5 (3) has 3, row6 (7) has 7, row7 (5) has 5, row4 (8) has 8, row8 (6) has 6, row9 (9) has 9, row10 (2) has 2. Missing \( r=1 \). Let's check column7 (1): \( r \times 1 = r \), so row with \( r=1 \) would have column7=1. Let's see column7: row2=4, row5=3, row6=7, row7=5, row4=8, row8=6, row9=9, row10=2. So missing row \( r=1 \), column7=1. Then \( r=1 \), column1: \( 1 \times 5 = 5 \), column2: \( 1 \times 8 = 8 \), column3: \( 1 \times 4 = 4 \), column4: \( 1 \times 9 = 9 \), column5: \( 1 \times 7 = 7 \), column6: \( 1 \times 3 = 3 \), column7: \( 1 \times 1 = 1 \), column8: \( 1 \times 2 = 2 \), column9: \( 1 \times 6 = 6 \).

Now, let's fill all cells:

| \( r \) \ \( c \) | 5 (c1) | 8 (c2) | 4 (c3) | 9 (c4) | 7 (c5) | 3 (c6) | 1 (c7) | 2 (c8) | 6 (c9) |
|-------------------|--------|--------|--------|--------|--------|--------|--------|--------|--------|
| 1 | 5 | 8 | 4 | 9 | 7 | 3 | 1 | 2 | 6 |
| 4 | 20 | 32 | 16 | 36 | 28 | 12 | 4 | 8 | 24 |
| 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: The initial table's row and column labels might be slightly different, but the key is using \( \text{product} = r \times c \) to find missing values by identifying \( r \) or \( c \) from known products.)

Answer:

The completed multiplication table (with rows as multipliers and columns as multiplicands, where column 1: 5, column 2: 8, column 3: 4, column 4: 9, column 5: 7, column 6: 3, column 7: 1, column 8: 2, column 9: 6) is filled by calculating each cell as \( r \times c \) for the respective row (\( r \)) and column (\( c \)) values. The final table has all cells filled with the correct products of their row and column numbers.