Bases: all problems

In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$ representation of $2013$ end in the digit $3$?

Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$?

For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of } 12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.

A positive integer $N$ has base-eleven representation $\underline{a}\,\underline{b}\,\underline{c}$ and base-eight representation $\underline{1}\,\underline{b}\,\underline{c}\,\underline{a}$, where $a$, $b$, and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.

For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?

Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\underline{32d}$ in base $n$ equals $263$, and the value of the numeral $\underline{324}$ in base $n$ equals the value of the numeral $\underline{11d1}$ in base six. What is $n + d$?

Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\underline{32d}$ in base $n$ equals $263$, and the value of the numeral $\underline{324}$ in base $n$ equals the value of the numeral $\underline{11d1}$ in base six. What is $n + d$?

For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$?

In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A + B$ represent number bases:

What is $A + B$?

The number $n$ can be written in base $14$ as $\underline{a}\,\underline{b}\,\underline{c}$, can be written in base $15$ as $\underline{a}\,\underline{c}\,\underline{b}$, and can be written in base $6$ as $\underline{a}\,\underline{c}\,\underline{a}\,\underline{c}$, where $a > 0$. Find the base-10 representation of $n$.

Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-$7$ representation of $N$ form a base-$10$ number that is twice $N$. For example, $51$ is a 7-10 double because its base-$7$ representation is $102$. What is the largest 7-10 double?

Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $N$ is divided by $1000$.

A rational number written in base eight is $\underline{a}\,\underline{b}\,.\,\underline{c}\,\underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b}\,\underline{b}\,.\,\underline{b}\,\underline{a}$. Find the base-ten number $\underline{a}\,\underline{b}\,\underline{c}$.

Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.

The number $n$ can be written in base $14$ as $\underline{a}\,\underline{b}\,\underline{c}$, can be written in base $15$ as $\underline{a}\,\underline{c}\,\underline{b}$, and can be written in base $6$ as $\underline{a}\,\underline{c}\,\underline{a}\,\underline{c}$, where $a > 0$. Find the base-10 representation of $n$.

Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_b$ is a perfect square and the base-$b$ integer $27_b$ is a perfect cube.

A positive integer $N$ has base-eleven representation $\underline{a}\,\underline{b}\,\underline{c}$ and base-eight representation $\underline{1}\,\underline{b}\,\underline{c}\,\underline{a}$, where $a$, $b$, and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.

There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1 \le k \le r$) with each $a_k$ either $1$ or $-1$ such that

Find $n_1 + n_2 + \cdots + n_r$.

For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of } 12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.

Bernardo chooses a three-digit positive integer $N$ and writes both its base-$5$ and base-$6$ representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-$10$ integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $10{,}444$ and $3{,}245$, and LeRoy obtains the sum $S = 13{,}689$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $2N$?