All Questions ⁺⁰236048 Questions

0

15

0

+1521

Number Theory

Write the following expression as a fraction in lowest terms:
2 + \cfrac{3}{3 + \cfrac{3}{3 + 3}}.

blackpanther Jul 16, 2024

0

16

1

+1521

Number Theory

Compute
4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4}}}}

blackpanther Jul 16, 2024

0

12

0

+1521

Number Theory

In base $b,$ the square of the number $17_b$ is $211_b$. Find the base $b$.

blackpanther Jul 16, 2024

0

22

0

+893

Number Theory

Let $S$ be the set of numbers of the form
n(n + 1)(n + 3)(n + 4)(n + 8)(n + 10)(n + 12)
where $n$ is any positive integer.
What is the GCD of the elements of $S$?

AnswerscorrectIy Jul 16, 2024

0

9

0

+893

Number Theory

There is a single sequence of integers $a_2,$ $a_3,$ $a_4,$ $a_5,$ $a_6,$ $a_7$ such that
\frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!} = \frac{3}{128}
and $0 \le a_i < i$ for $i = 2,$ read more ..

AnswerscorrectIy Jul 16, 2024

0

22

0

+893

Number Theory

Find the unique triple $(x,y,z)$ of positive integers such that $x < y < z$ and
\frac{1}{x} - \frac{1}{xy} - \frac{1}{xyz} = \frac{1}{3}

AnswerscorrectIy Jul 16, 2024

0

15

0

+893

Number Theory

Find the largest positive integer $n$ such that $n^3$ divides 15.

AnswerscorrectIy Jul 16, 2024

0

21

0

+893

Number Theory

When expanded as a decimal, the fraction \frac{1}{977} has a repetend (the repeating part of the decimal) that begins right after the decimal point, and is $976$ digits long. Find the first three digits after the decimal point.

AnswerscorrectIy Jul 16, 2024

0

14

0

+893

Number Theory

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

AnswerscorrectIy Jul 16, 2024

0

11

0

+893

Number Theory

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 ..

AnswerscorrectIy Jul 16, 2024

0

18

0

+893

Number Theory

A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest terrific positive integer?

AnswerscorrectIy Jul 16, 2024

0

10

1

+842

Number Theory

Express 2.\overline{57} as a base 10 fraction in reduced form.

RedDragonl Jul 16, 2024

0

14

0

+842

Number Theory

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 ..

RedDragonl Jul 16, 2024

0

19

1

+842

Number Theory

Express 0.11012 as a base 10 fraction in reduced form.

RedDragonl Jul 16, 2024

0

17

1

+842

Number Theory

What is the largest positive integer $n$ such that $1457$, $1797$, $709$, $15$, $24$, $197$, $428$ all leave the same remainder when divided by $n$?

RedDragonl Jul 16, 2024

-1

11

0

+1839

Probability

The following cards are split into three piles at random, so that every pile contains the same number of cards. What is the probability that every pile contains an Ace?
Ace of spades
Ace of hearts
Ace of diamonds read more ..

ABJeIIy Jul 16, 2024

-1

18

1

+1839

Number Theory

The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
10011010011ABC110_2.
Compute the ordered triple of digits $(A,B,C)$.

ABJeIIy Jul 16, 2024

0

19

0

+893

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

AnswerscorrectIy Jul 16, 2024

0

17

0

+893

Algebra

Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.

AnswerscorrectIy Jul 16, 2024

0

20

1

+893

Algebra

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

AnswerscorrectIy Jul 16, 2024

0

14

1

+893

Algebra

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

AnswerscorrectIy Jul 16, 2024

0

15

0

+1911

Geometry

In trapezoid $PQRS,$ $\overline{PQ} \parallel \overline{RS}$. Let $X$ be the intersection of diagonals $\overline{PR}$ and $\overline{QS}$. The area of triangle $PQX$ is $2,$ and the area of triangle $RSX$ is $2.$ Find the area of trapezoid read more ..

tomtom Jul 16, 2024

0

9

1

+1911

Geometry

In triangle ABC, angle ACB = 90 degrees. Let H be the foot of the altitude from C to side AB.
If BC = 1 and AH = 2, find CH.

tomtom Jul 16, 2024

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