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All Questions
^{+0}
234583 Questions
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4
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+480
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
0
6
1
+1694
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?</
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ABJeIIy
Jun 22, 2024
0
3
1
+1694
Number Theory
How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$
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ABJeIIy
Jun 22, 2024
0
7
0
+1694
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
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ABJeIIy
Jun 22, 2024
0
3
1
+1694
Number Theory
What is the smallest prime divisor of $5^{19} + 7^{13} + 23$?
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ABJeIIy
Jun 22, 2024
0
8
0
+1694
Number Theory
How many bases $b \ge 2$ are there such that $100_b + 1_b$ is prime?
ABJeIIy
Jun 22, 2024
0
15
0
+1039
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
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learnmgcat
Jun 22, 2024
0
6
0
+1039
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
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learnmgcat
Jun 22, 2024
0
6
0
+1039
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,
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learnmgcat
Jun 22, 2024
0
6
0
+1039
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
0
8
0
+1039
Number Theory
Find the greatest prime divisor of the value of the arithmetic series
5 + 6 + 7 + \dots + 135.
learnmgcat
Jun 22, 2024
0
9
1
+480
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).
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gnistory
Jun 22, 2024
0
9
1
+480
Algebra
Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 11x - 6. Find u/v^2 + v/u^2.
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gnistory
Jun 22, 2024
0
11
1
+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
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derpmcfearson
Jun 22, 2024
0
16
0
+480
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
-3
7
0
+165
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 $\angle STU = 45^\circ$,
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BRAlNBOLT
Jun 21, 2024
-3
9
0
+165
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 $AB = 20$, $AC = 20$,
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BRAlNBOLT
Jun 21, 2024
0
9
1
+4
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
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MEdge
Jun 21, 2024
0
12
0
+480
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
-3
9
0
+165
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
11
0
+165
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
-2
15
0
+165
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
0
7
1
+1341
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?
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kittykat
Jun 21, 2024
0
4
1
+1341
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 ..
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kittykat
Jun 21, 2024
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