All Questions ⁺⁰236320 Questions

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Euclidean Algorithm

As shown in class, the Euclidean algorithm can be used to find solutions to equations of the form
\[ax + by = c.\]
Use the Euclidean algorithm to find integers $x$ and $y$ such that $266x + 357y = 1,$ with the smallest possible read more ..

gnistory Jun 22, 2024

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+1839

Number Theory

Starting at the same time, Bill rings a bell every $36$ seconds, and Wendy blows a whistle every $90$ seconds, and Sam blows a horn every $385$ seconds. How many minutes will it be before both Bill and Wendy and Sam simultaneously make a sound?</ read more ..

ABJeIIy Jun 22, 2024

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+1839

Number Theory

How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$

ABJeIIy Jun 22, 2024

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+1839

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 ..

ABJeIIy Jun 22, 2024

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Number Theory

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

ABJeIIy Jun 22, 2024

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Number Theory

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

ABJeIIy Jun 22, 2024

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21

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+1379

Number Theory

What are the first $5$ digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{17}$ is written in base $16$?
(Express your answer as a five digit hexadecimal number. You do not need to read more ..

learnmgcat Jun 22, 2024

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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 ..

learnmgcat Jun 22, 2024

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Number Theory

What is 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 ..

learnmgcat Jun 22, 2024

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+1379

Number Theory

Find the smallest positive integer $B$ so that when we express the decimal number $250$ as a base $B$ number, we still get a $2$-digit number.

learnmgcat Jun 22, 2024

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+1379

Number Theory

Find the greatest prime divisor of the value of the arithmetic series
5 + 6 + 7 + \dots + 135.

learnmgcat Jun 22, 2024

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Algebra

Let a and b be the solutions to 7x^2 + x - 5 = -3x^2 - 9x - 4. Compute (a - 4)/(b - 4) + (b - 4)/(a - 4).

gnistory Jun 22, 2024

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Algebra

Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 11x - 6. Find u/v^2 + v/u^2.

gnistory Jun 22, 2024

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+22

Please help. Thanks

Kenneth and Marshall started cycling from the same starting point along a track. Both started cycling at the same time and they did not change their speeds throughout. Kenneth reached the end of the track in 2 hours: Marshall covered only 4/5 of the read more ..

derpmcfearson Jun 22, 2024

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+845

Algebra

Let a and b be the solutions to 5x^2 - 11x + 4 = 4x^2 - 17x + 6. Find 1/a^3 + 1/b^3.

gnistory Jun 22, 2024

Jun 21, 2024

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Geometry

Let $M$, $N$, and $P$ be the midpoints of sides $\overline{TU}$, $\overline{US}$, and $\overline{ST}$ of triangle $STU$, respectively. Let $\overline{UZ}$ be an altitude of the triangle. If $\angle TSU = 60^\circ$ and $\angle STU = 45^\circ$, read more ..

BRAlNBOLT Jun 21, 2024

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Geometry

Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $BET$ if $AB = 20$, $AC = 20$, read more ..

BRAlNBOLT Jun 21, 2024

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Quick Question

It's karaoke night at the local bar. A group of 5 friends, who seem to have a knack for mathematics and a love for K-pop, walk in and order 3 beers each. One of them mentions they're celebrating a birthday with Chinese food later. Assuming the bartender read more ..

MEdge Jun 21, 2024

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+845

Geometry

In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.

gnistory Jun 21, 2024

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Coordinates

Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.

BRAlNBOLT Jun 21, 2024

-3

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+219

Coordinates

Points $A,$ $B,$ and $C$ are given in the coordinate plane. There exists a point $Q$ and a constant $k$ such that for any point $P$,
PA^2 + PB^2 + PC^2 = 3PQ^2 + k.
If $A = (7,-11),$ $B = (10,13),$ and $C = (18,-22)$, then find the constant read more ..

BRAlNBOLT Jun 21, 2024

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Algebra

Let $x$ and $y$ be real numbers. If $x$ and $y$ satisfy
x^2 + y^2 = 4x - 8y + 17x - 5y + 25,
then find the largest possible value of $x.$ Give your answer in exact form using radicals, simplified as far as possible.

BRAlNBOLT Jun 21, 2024

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Counting

I have $3$ different mathematics textbooks, $2$ different psychology textbooks, and $2$ different chemistry textbooks. In how many ways can I place the $7$ textbooks on a bookshelf, in a row?

kittykat Jun 21, 2024

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Probability

Raina places the following balls into a bag. She then draws three balls out of the bag, one at a time, without replacement. What is the probability that the colors of the balls alternate?
There are 3 orange balls and 2 purple read more ..

kittykat Jun 21, 2024

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