Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.
Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Find the $1000^{th}$ number in $S$ (in base $10$).
When the integers 1 through 7 are written in base two, what fraction of the digits are 1s? Express your answer as a common fraction.
In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A+43_B=69_{A+B}.\] What is $A+B$?
The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
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 $n$?
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 $3$?
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 $2N$?
If $12_3$ + $12_5$ + $12_7$ + $12_9$ + $12_x$ = $101110_2$ , what is the value of $x$, the base of the fifth term?
Show that any positive integer can be expressed as a sum of integers which are some power of 3.
A binary palindrome is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeros are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?
What is the minimal number of masses required in order to measure any weight between 1 and $n$ grams. Note that a mass can be put on either sides of the balance.
Find the number of ending zeros of $2014!$ in base 9. Give your answer in base 9.
What is the $22^{nd}$ positive integer $n$ such that $22^n$ ends in a $2$?
What is the largest positive integer $n$ less than $10,000$ such that in base 4, $n$ and $3n$ have the same number of digits; in base 8, $n$ and $7n$ have the same number of digits; and in base 16, $n$ and $15n$ have the same number of digits? Express your answer in base 10.
A base-10 three digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?
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 $0$.
Let $\mathbb{S}$ be the set of integers between $1$ and $2^{40}$ that contain two $1$s when
written in base $2$. What is the probability that a random integer from $\mathbb{S}$ is divisible by $9$?
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k=0.232323\cdots_k$. What is $k$?