Practice (91)

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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?

How many positive two-digit integers have exactly $8$ positive factors?

What fraction of the first 100 triangular numbers are evenly divisible by 7? Express your answer as a common fraction.

The sum of three primes is 125. The difference between the largest and the smallest is 50. What is the largest possible median of these three prime numbers?

We have a set of numbers {1, 2, 3, 4, 5} and we take products of three different numbers. We must find now many pairs of relatively prime numbers there are.

What is the largest five-digit integer such that the product of the digits is $2520$?

For how many two-element subsets {$a,b$} of the set {$1, 2, 3, \cdots, 36$} is the product of $ab$ a perfect square?

What percent of the positive integers $\le$ 36 are factors of 36?

$S$ and $T$ are both two-digit integers less than 80. Each number is divisible by 3. $T$ is also divisible by 7. $S$ is a perfect square. $S + T$ is a multiple of 11, so what is the value of $T$?

How many positive integers not exceeding $2000$ have an odd number of factors?

Find the smallest positive integer $n$ such that 20 divides $15n$ and 15 divides $20n$.

Find the smallest positive integer $N$ such that $2N$ is a perfect square and $3N$ is a perfect cube.

All positive integers relatively prime to 2015 are written in increasing order. Let the twentieth number be $p$. The value of $\frac{2015}{p}\u2212 1$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $a + b$.

For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?

Find the least composite positive integer that is not divisible by any of 3, 4, and 5.

Let the positive divisors of $n$ be $d_1, d_2, \dots$ in increasing order. If $d_6 = 35$, determine the minimum possible value of $n$.

Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank?

Compute the smallest positive integer with at least four two-digit positive divisors.

Find the maximum possible value of the greatest common divisor of $MOO$ and $MOOSE$, given that $S$, $O$, $M$, and $E$ are some nonzero digits. (The digits $S$, $O$, $M$, and $E$ are not necessarily pairwise distinct.)

Can a five-digit number consisting of 5 distinct even digits a perfect square?

Find the largest multiple of 99 among the nine-digit integers, whose digits are all distinct.

How many positive integers less than $1998$ are relatively prime to $1547$?

For a positive integer m, we define $m$ as a $\textit{factorial}$ number if and only if there exists a positive integer $k$ for which $m = k\cdot(k - 1)\cdots 2\cdot 1$. We define a positive integer $n$ as a $\textit{Thai}$ number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers. What is the sum of the five smallest $\textit{Thai}$ numbers?

Express the golden ratio using a continued fraction.

Find the rational number $p/q$ closest to $\sqrt{\pi}$ wich $q \le 25$.