Elementary maths in numeric bases

Base change related problems make for good puzzles. Here’s a shot list I have worked out, problems were made via GPT using some my ideas so… well the sources are lost to Mr. Altman’s copyright erasure machine.

all that’s needed

• A numeral written in base $b$ can be expanded using powers of $b$:

  $$(pqr.xyz{)}_b=p{b}^2+qb+r+\frac{x}{b}+\frac{y}{b^2}+\frac{z}{b^3}.$$

• If a digit $d$ appears, then the base must satisfy $b>d.$

problems

1. The point $(5,(31{)}_b)$ lies on the parabola

   $y=x^2.$

   Find $b$.

   [Hint: Express $(31{)}_b$ as a polynomial in $b$.]

2. In base $b$,

   $(46{)}_b+(37{)}_b=(105{)}_b.$

   Find $b$.

   [Hint: Examine the units column and the carry. Ugly solution would be to expand it as polynomial]

3. The roots of

   $x^2-(10{)}_{b}x+(31{)}_b=0$

   are $5$ and $8$. Find $b$.

   [Hint: Use the sum and product of the roots.]

4. Determine all valid bases $b$ for which

   $(11{)}_b^2=(121{)}_b.$

   [Hint: Determine whether the equation selects one base or is an identity.]

5. Find all valid bases $b$ for which

   $(144{)}_b$

   is a perfect square.

   [Hint: Expand the numeral and complete the square.]

6. For which valid bases $b$ is

   $(22{)}_b$

   a prime number?

   [Hint: Factor its value as an expression in $b$.]

7. In base $b$,

   $(0.\overline{3}{)}_b=\frac{1}{2}.$

   Find $b$.

   [Hint: Write the repeating expansion as an infinite geometric series.]

8. Find every valid base $b$ for which

   $(1234321{)}_b$

   is divisible by $b-1$.

   [Hint: Work modulo $b-1$, where $b\equiv 1$.]

9. Prove that

   $(123321{)}_b$

   is divisible by $b+1$ in every valid base.

   [Hint: Work modulo $b+1$, where $b\equiv -1$.]

10. For which valid integer bases $b$ is

    $(123{)}_b$

    divisible by $3$?

    [Hint: Consider each possible value of $b(\mathrm{mod}3)$.]

11. For which integer bases $b\ge 2$ does

    $\frac{1}{6}$

    have a terminating base-$b$ expansion?

    [Hint: A terminating expansion can be written with denominator $b^n$ for some $n$.]