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 Question from David, a student: In what base is 3x3= 10, 3x3=11, 3x3 = 12? is there a fast way to see this or do I have to create multiplication tables until I find the right one?

David,

For a two digit number in base ten the leftmost digit is the tens digit and the rightmost digit is the units digit. Thus 43 is 4 tens and 3 units. For a two digit number in base b the leftmost digit is the b digit and the rightmost digit is the units digit. Thus 43b is 4 bs and 3 units. So if b is 8 then 438 is 4 eights and 3 units, that is thirty-two plus three which is thirty-five.

This means that 10b is one b and zero units. Thus the value of 10b is the value of the base b. Hence if b = 8 as above then 108 is eight.

Hence in your question above 10b is the value of the base, 11b is the value of the base, plus one and 12b is the value of the base, plus 2. So if the base is 8 then 108 is eight, 118 is nine and 128 is ten.

I hope this helps,
Penny

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