Question

1-1 homework name complete the multiplication table.

Answer:

To complete the multiplication table, we use the rule that each cell in row \( r \) and column \( c \) is \( r \times c \). Let's fill in the missing cells step - by - step:

Step 1: Analyze the row and column headers

The rows are labeled with numbers \( 1,2,3,4,5,6,7,8,9 \) (let's assume the row labels are on the left - most column) and the columns are also labeled with numbers \( 1,2,3,4,5,6,7,8,9 \) (let's assume the column labels are on the top - most row). For a cell at the intersection of row \( i \) and column \( j \), the value should be \( i\times j \).

Step 2: Fill in the missing cells

• Row 1 (all columns): Since \( 1\times j=j \) for any column \( j \). So the cells in row 1 (from column 1 to 9) are \( 1,2,3,4,5,6,7,8,9 \) respectively.
• Row 2 (column 3): \( 2\times3 = 6 \); (column 4): \( 2\times4=8 \); (column 5): \( 2\times5 = 10 \); (column 6): \( 2\times6=12 \); (column 7): \( 2\times7 = 14 \); (column 8): \( 2\times8=16 \); (column 9): \( 2\times9 = 18 \)
• Row 3 (column 2): \( 3\times2=6 \); (column 3): \( 3\times3 = 9 \); (column 4): \( 3\times4=12 \); (column 5): \( 3\times5 = 15 \); (column 6): \( 3\times6=18 \); (column 7): \( 3\times7 = 21 \); (column 8): \( 3\times8=24 \); (column 9): \( 3\times9 = 27 \)
• Row 4 (column 1): \( 4\times1 = 4 \); (column 2): \( 4\times2=8 \); (column 3): \( 4\times3 = 12 \); (column 4): \( 4\times4=16 \); (column 5): \( 4\times5 = 20 \); (column 6): \( 4\times6=24 \); (column 7): \( 4\times7 = 28 \); (column 8): \( 4\times8=32 \); (column 9): \( 4\times9 = 36 \)
• Row 5 (column 2): \( 5\times2 = 10 \); (column 3): \( 5\times3=15 \); (column 4): \( 5\times4 = 20 \); (column 5): \( 5\times5=25 \); (column 6): \( 5\times6 = 30 \); (column 7): \( 5\times7=35 \); (column 8): \( 5\times8 = 40 \); (column 9): \( 5\times9=45 \)
• Row 6 (column 1): \( 6\times1=6 \); (column 2): \( 6\times2 = 12 \); (column 3): \( 6\times3=18 \); (column 4): \( 6\times4 = 24 \); (column 5): \( 6\times5=30 \); (column 6): \( 6\times6 = 36 \); (column 7): \( 6\times7=42 \); (column 8): \( 6\times8 = 48 \); (column 9): \( 6\times9=54 \)
• Row 7 (column 1): \( 7\times1 = 7 \); (column 2): \( 7\times2=14 \); (column 3): \( 7\times3 = 21 \); (column 4): \( 7\times4=28 \); (column 5): \( 7\times5 = 35 \); (column 6): \( 7\times6=42 \); (column 7): \( 7\times7 = 49 \); (column 8): \( 7\times8=56 \); (column 9): \( 7\times9 = 63 \)
• Row 8 (column 1): \( 8\times1=8 \); (column 2): \( 8\times2 = 16 \); (column 3): \( 8\times3=24 \); (column 4): \( 8\times4 = 32 \); (column 5): \( 8\times5=40 \); (column 6): \( 8\times6 = 48 \); (column 7): \( 8\times7=56 \); (column 8): \( 8\times8 = 64 \); (column 9): \( 8\times9=72 \)
• Row 9 (column 1): \( 9\times1=9 \); (column 2): \( 9\times2 = 18 \); (column 3): \( 9\times3=27 \); (column 4): \( 9\times4 = 36 \); (column 5): \( 9\times5=45 \); (column 6): \( 9\times6 = 54 \); (column 7): \( 9\times7=63 \); (column 8): \( 9\times8 = 72 \); (column 9): \( 9\times9=81 \)

The completed multiplication table (showing some of the key fills based on the given partial table) is as follows:

\(\boldsymbol{\cdot}\)	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

If we consider the original table's partial fills, for example, in row 2, column 4 should be \( 2\times4 = 8 \), row 3, column 2 should be \( 3\times2=6 \) etc. and we fill the table using the multiplication rule \( \text{cell}(i,j)=i\times j \) where \( i \) is the row number and \( j \) is the column number.