All Questions ⁺⁰234656 Questions

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Algebra

Find the minimum value of \frac{x^2}{x - 1 + x^3} for x > 1

BRAlNBOLT Jul 18, 2024

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6

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Algebra

Find all values of t such that floor(t) = 3t + 4 - t^2. If you find more than one value, then list the values you find in increasing order, separated by commas.

BRAlNBOLT Jul 18, 2024

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Algebra

Let
f(x) = \left\lfloor\frac{2 - 3x}{3x + 1}\right\rfloor.
Evaluate f(1)+f(2) + f(3) + \dots + f(999)+f(1000). (This sum has 1000 terms, one for the result when we input each integer from 1 to 1000 into f.)

BRAlNBOLT Jul 18, 2024

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+820

Number Theory

A four-digit hexadecimal integer is written on a napkin such that the units digit is illegible. The first three digits are 2, $F$, and 1. If the integer is a multiple of $19_{10}$, what is the units digit?

Hi6942O Jul 18, 2024

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+820

Number Theory

The numbers $24^2 = 576$ and $56^2 = 3136$ are examples of perfect squares that have a units digits of $6.$
If the units digit of a perfect square is $5,$ then what are the possible values of the tens digit?

Hi6942O Jul 18, 2024

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2

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+820

Number Theory

Which of the residues 0, 1, 2, ..., 11 satisfy the congruence 3x = 1 mod 12?

Hi6942O Jul 18, 2024

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+820

Number Theory

Which of the residues 0, 1, 2, 3, 4 satisfy the congruence x^5 = 0 mod 5?

Hi6942O Jul 18, 2024

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5

0

+779

Counting

In how many ways can the numbers 1, 2, 3, 4, 5, 6 be arranged in a row, so that the product of any two adjacent numbers is at least 5?

eramsby1O1O Jul 18, 2024

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7

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+779

Counting

Find the number of $7$-digit numbers, where the sum of the digits is divisible by $11.$

eramsby1O1O Jul 18, 2024

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Number Theory

When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.

When read more ..

BRAlNBOLT Jul 18, 2024

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Number Theory

What is the smallest prime divisor of $5^{19} + 7^{13} + 23$?

BRAlNBOLT Jul 18, 2024

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+165

Number Theory

How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?

BRAlNBOLT Jul 18, 2024

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5

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+165

Number Theory

A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?

BRAlNBOLT Jul 18, 2024

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7

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Number Theory

The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$
What is the smallest positive integer that has exactly $2$ perfect square divisors?

BRAlNBOLT Jul 18, 2024

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4

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Number Theory

For a positive integer $n$, $\phi(n)$ denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$.
What is $\phi(1200)$?

BRAlNBOLT Jul 18, 2024

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1

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I need help

Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$.
Be sure to include complete explanations with your answer, using complete sentences. Imagine you were going to show your solution to a classmate, read more ..

BRAlNBOLT Jul 18, 2024

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Help me

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest number of primes that could divide a terrific positive integer?

NotThatSmart ●

BRAlNBOLT Jul 18, 2024

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I need help with this

You have a total supply of $1000$ pieces of candy, and an empty vat. You also have a machine that can add exactly $5$ pieces of candy per scoop to the vat, and another machine that can remove exactly $3$ pieces of candy with a different scoop from read more ..

BRAlNBOLT Jul 18, 2024

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I have an idea on this question, but not sure about it

Let $N$ be a positive integer. The number $N$ has three digits when expressed in base $7$. When the number $N$ is expressed in base $12$, it has the same three digits, in reverse order. What is $N$? (Express your answer in decim read more ..

BRAINBOLT Jul 18, 2024

Jul 17, 2024

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5

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+868

Number Theory

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

NotThatSmart ●

Rangcr897 Jul 17, 2024

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3

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+868

Number Theory

The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
101010010101000101A10001B001010C100011_2.
Compute $A,$ $B,$ and $C$.

Rangcr897 Jul 17, 2024

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5

+868

Number Theory

A school orders $99$ textbooks, all for the same price. When the bill for the total order comes, the first and last digits are obscured. What are the missing digits?
_18,486.7_

Bosco ●●●●●

Rangcr897 Jul 17, 2024

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6

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+868

Number Theory

The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$. Find the remainder when $F_{1000}$ is divided by $17$.

Rangcr897 Jul 17, 2024

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+868

Number Theory

How many of the 1000 smallest positive integers are congruent to 7 modulo 29?

Rangcr897 Jul 17, 2024

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