All Questions ⁺⁰237293 Questions

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33

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+965

Geometry

In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$. Given $AB = 14$, $AY = 14$, and $CY = 8$, find $BZ$.

magenta Jul 5, 2024

0

38

0

+965

Geometry

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $AX = 5,$ and $CX = 5$. What is $BX$?

magenta Jul 5, 2024

0

35

0

+965

Geometry

Find the area of triangle $ABC$ if $AB = 6,$ $BC = 8,$ and $\angle ACB = 30^\circ$.

magenta Jul 5, 2024

0

20

2

+8

I require help! Algebra question

A parabola with equation ax^2+bx+c=y has a vertical line of symmetry at X=2 and goes through the two points (1,1) and (4,-7). The quadratic ax^2+bx+c has two real roots. The greater root is . What is n?

Iamrockingtoday Jul 5, 2024

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22

1

+613

Number Theory

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

Pythagorearn Jul 5, 2024

0

34

0

+613

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 ..

Pythagorearn Jul 5, 2024

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27

1

+613

Number Theory

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

Pythagorearn Jul 5, 2024

0

22

1

+613

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.

Pythagorearn Jul 5, 2024

+1

21

1

+821

Number Theory

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

cooIcooIcooI17 Jul 5, 2024

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25

0

+821

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}

cooIcooIcooI17 Jul 5, 2024

0

27

0

+821

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 ..

cooIcooIcooI17 Jul 5, 2024

0

42

0

+821

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 $S$?

cooIcooIcooI17 Jul 5, 2024

0

21

1

+972

Number Theory

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

Rangcr897 Jul 5, 2024

0

29

0

+972

Number Theory

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

Rangcr897 Jul 5, 2024

0

27

0

+972

Number Theory

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

Rangcr897 Jul 5, 2024

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41

0

+972

Number Theory

Find an integer $x$ such that $0 \leq x < 205$ and $x^{89} \equiv 17 \pmod{5}$.

Rangcr897 Jul 5, 2024

0

27

0

+1501

Number Theory

The House of Lilliput is using RSA encryption to receive secret messages from all the realms. They have published their public encoding exponent $e = 8$ and their public modulus $M = pq = 6$.
Break the code: Find their secret decoding read more ..

learnmgcat Jul 5, 2024

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30

0

+1501

Number Theory

Carlos wants to send Cecil a message encrypted with RSA. Cecil has published his public encryption exponent $3$ and his public modulus $3$. When Carlos encrypts the number $3$ with this system, what is the result?

learnmgcat Jul 5, 2024

0

30

0

+1501

Number Theory

An eccentric baseball card collector wants to distribute her collection among her descendants. If she divided her cards among her $13$ great-great-grandchildren, there would be $8$ cards left over. If she divided her cards among her $10$ great-grandchildren, read more ..

learnmgcat Jul 5, 2024

Jul 4, 2024

0

17

1

+1501

Number Theory

When $N$ is divided by $10$, the remainder is $a$. When $N$ is divided by $2$, the remainder is $b$. What is $N$ modulo $20$, in terms of $a$ and $b$?

learnmgcat Jul 4, 2024

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34

0

+1501

Number Theory

John and Ling start their new jobs on the same day. John's schedule is six workdays followed by one day off. Ling's schedule is seven workdays followed by one day off. On how many days during their first year of work ($365$ days) read more ..

learnmgcat Jul 4, 2024

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26

1

+1501

Number Theory

Brandon spends his afternoon picking apples from an orchard. He notices that if he groups the apples he picked by fives, six, or sevens, then he ends up with $1$ left over. If he groups by eights, he has $1$ left over. If Brandon read more ..

learnmgcat Jul 4, 2024

0

29

0

+1501

Number Theory

Find the number of solutions to
N &\equiv 2 \pmod{6}, \\
N &\equiv 2 \pmod{7}, \\
N &\equiv 2 \pmod{8}
in the interval 0 \le N < 1000.

learnmgcat Jul 4, 2024

0

29

0

+1501

Number Theory

Find the number of solutions to
N &\equiv 2 \pmod{5}, \\
N &\equiv 2 \pmod{6}, \\
N &\equiv 2 \pmod{7}
in the interval 0 \le N < 1000.

learnmgcat Jul 4, 2024

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