A rectangle with positive integer side lengths in $\text{cm}$ has area $A$ $\text{cm}^2$ and perimeter $P$ $\text{cm}$. Which of the following numbers cannot equal $A+P$?
Among the positive integers less than $100$, each of whose digits is a prime number, one is selected at random. What is the probability that the selected number is prime?
Let $n$ be a positive integer greater than $4$ such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?
If integer $a$, $b$, $c$, and $d$ satisfy $ad-bc=1$. Prove $a+b$ and $c+d$ are relatively prime.
Prove for any positive integer $n$, the fraction $\frac{21n+4}{14n+3}$ cannot be further simplified.
Prove: there exists a rational number $\frac{c}{d}$, where $d<1000$, such that $$\Big[k\cdot\frac{c}{d}\Big]=\Big[k\cdot\frac{73}{100}\Big]$$ holds for every positive integer $k$ that is less than 1000. Here $\Big[x\Big]$ denotes the largest integer that is not exceeding $x$.
Which of the following number is a perfect square?
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor?
What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?
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 $3$?
Three positive integers are each greater than $1$, have a product of $27000$, and are pairwise relatively prime. What is their sum?
The number $2013$ is expressed in the form
$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$ ?
A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numbers in the set $\{ 2010,2011,2012,\dotsc,2019 \}$ are nice?
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 $2N$?
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of even integers among the 6 integers?
How many ordered pairs of positive integers (M,N) satisfy the equation $\frac {M}{6}$ = $\frac{6}{N}$
For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.
Let $M$ be the product of any four consecutive positive integers. Prove $M+1$ must be a perfect square.
Prove $(n^2 -1)$ and $n$ are relatively prime.
Prove: if $(2^n+1)$ is a prime number, then $n$ must be some power of 2.
If $2^n-1$ is a prime number, prove $n$ must be a prime number too.
A majority of the 30 students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents?