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If $A=(b-1) (b-1) (b-1)$ and $B=(b-1) (b-1)$ are written in base $b$, what is $A\times B$ in base $b$?

I've tried developing $A \times B$ in decimal base but couldn't get this product back in base $b$.

Beside this, is there a general procedure to compute product in non decimal base.

Thanks in advance for your help,

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    By $(b-1)(b-1)(b-1)$ are you meaning the product or trying to write a three digit numbre? E.g. If $b=10$ do you mean $9^3=729$ or $999$? – 2017-01-13
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    999, Im writing in base b – 2017-01-13

3 Answers 3

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I assume you mean "what's the product of two numbers $n$ and $m$ whose digits are all equal to the largest digit you can write in that base?" (e.g. $666_7$ and $66_7$ - in base $7$)?

Hint: what are the digits of $(n+1)\times(m+1) = nm + ((n+1) +(m+1) -1)$?

In general, multiplication in all bases is the same as in base $10$: multiply the digits $1$ by $1$ as you'd do in base 10, but obviously with a different multiplication table, with the usual rules for carryover.

3

Your two numbers as the largest possible three digit and largest possible two digit number written in base $b$. I.e. they are $1000_b-1_b$ and $100_b-1_b$. Algebraically these are equal to $b^3-1$ and $b^2-1$ respectively. As such their product will be given by:

$$(b^3-1)(b^2-1)=b^5-b^3-b^2+1$$

This however has negatives in it so its hard to read it as a base $b$ number. Lets manipulate until it is all positive values multiplied by powers of $b$.

$$=(b-1+1)b^4-b^3-b^2+1$$

$$=(b-1)b^4+b^4-b^3-b^2+1$$

$$=(b-1)b^4+(b-2+2)b^3-b^3-b^2+1$$

$$=(b-1)b^4+(b-2)b^3+2b^3-b^3-b^2+1$$

$$=(b-1)b^4+(b-2)b^3+b^3-b^2+1$$

$$=(b-1)b^3+(b-2)b^3+(b-1+1)b^2-b^2+1$$

$$=(b-1)b^3+(b-2)b^3+(b-1)b^2+b^2-b^2+1$$

$$=(b-1)b^3+(b-2)b^3+(b-1)b^2+1$$

This is now a sum of positive numbers times a power of $b$ so we can read off the coefficients to give up the digits in base $b$. So the digits in your product are $b-1$, $b-2$, $b-1$, $0$, $1$. E.g. in base 10 it would be 98901, in base 4 it would be 32301, etc.

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Let throw away the usual meanings of $8,9,10$ and replace them with $10$ means $b$, $9$ means simply $b -1$ and $8$ means $b-2$. We assume $b \ge 3$ and so if we were dealing with base $7$ or base $32$ then $9$ and $8$ means "six" and "five" in base $7$, and $9$ and $8$ means "thirty-one" and "thirty" in base $32$.

Okay. So

$A = (b-1)(b-1)(b-1) = 999 = b^3 -1 = 1000 - 1$

$B = (b-1)(b-1) = 99 = b^2 - 1 = 100 - 1$

So $AB = (b^3 -1)(b^2 - 1) = (1000 -1)(100 - 1)$

$100000 - 1000 - 100 + 1 = $

$99000 - 100 + 1 = $

$98900 + 1 = 98901 = (b-1)(b-2)(b-1)01$

This will be true for all $b \ge 3$

So $b=3: 21201$

$b=4: 32301$

$b=5: 43401$.....

$b=9: 87801$

$b=10: 98901$

$b=11: A9A01$.....

$b =16: FEF01$, etc.