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 Question from sylvia, a student: I am having a difficult time trying to figure out how to fill in the multiplication table for the different bases. i don't know how to get the numbers.

Hi Sylvia,

I am going to use base seven.

In base seven there are seven digits, 0, 1, 2, 3, 4, 5, and 6.Hence my multiplication table will look like

0 1 2 3 4 5 6
0
1
2
3
4
5
6

Multiplication table base 7

The first row is easy since $0 \times b = 0$ for any number $b$ in any base.

0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1
2
3
4
5
6

Multiplication table base 7

Also multiplication is commutative, $a \times b = b \times a$ so once you complete a row you can fill in the corresponding column.

0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0
2 0
3 0
4 0
5 0
6 0

Multiplication table base 7

The second row and second column are also easy since $1 \times b = b \times 1$ for any $b$ in any base.

0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2
3 0 3
4 0 4
5 0 5
6 0 6

Multiplication table base 7

$2 \times 2 = 4 \mbox{ and } 2 \times 3 = 6$ so the next two entries in the third row are $4$ and $6$ but $2 \times 4$ is eight and I don't have a digit for eight. But eight is seven plus 1, that is one seven and one more so in base seven, eight is $11_7.$ Similarly $2 \times 5$ is ten which is seven plus three so in base seven, ten is $13_7.$ Likewise $2 \times 6$ is twelve which is seven plus five so in base seven, twelve is $15_7.$ Thus I can complete the second row and column.

0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 117  137 157
3 0 3 6
4 0 4 117
5 0 5 137
6 0 6 157

Multiplication table base 7

I will find one more entry, $5 \times 6.$ $5 \times 6$ is thirty. If you divide thirty by seven you get 4 with a remainder of 2. Thus thirty is four sevens and two more so in base seven, thirty is $42_7.$ Hence I get

0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 117  137 157
3 0 3 6
4 0 4 117
5 0 5 137        427
6 0 6 157      427

Multiplication table base 7

You can complete the table.

Write back if you need more assistance,
Penny

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