problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Suppose 5 different integers are randomly chosen from between 20 and 69, inclusive. What is the probability that they each have a different tens digit? | \frac{2500}{52969} |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | \frac{\sqrt{15}}{3} |
The graph of the function $y=g(x)$ is given. For all $x > 5$, it holds that $g(x) > 0.5$. The function $g(x)$ is defined as $g(x) = \frac{x^2}{Ax^2 + Bx + C}$ where $A$, $B$, and $C$ are integers. The vertical asymptotes of $g$ are at $x = -3$ and $x = 4$, and the horizontal asymptote is such that $y = 1/A < 1$. Find $... | -24 |
What is the largest factor of $130000$ that does not contain the digit $0$ or $5$ ? | 26 |
A shooter fires 5 shots in succession, hitting the target with scores of: $9.7$, $9.9$, $10.1$, $10.2$, $10.1$. The variance of this set of data is __________. | 0.032 |
The Greater Fourteen Basketball League has two divisions, each containing seven teams. Each team plays each of the other teams in its own division twice and every team in the other division twice. How many league games are scheduled? | 182 |
Vasya and Petya simultaneously started running from the starting point of a circular track in opposite directions at constant speeds. At some point, they met. Vasya completed a full lap and, continuing to run in the same direction, reached the point of their first meeting at the same moment Petya completed a full lap. ... | \frac{1+\sqrt{5}}{2} |
How many two-digit positive integers are congruent to 1 (mod 3)? | 30 |
On the grid shown, Jane starts at dot $A$. She tosses a fair coin to determine which way to move. If she tosses a head, she moves up one dot. If she tosses a tail, she moves right one dot. After four tosses of the coin, Jane will be at one of the dots $P, Q, R, S$, or $T$. What is the probability that Jane will be at d... | $\frac{3}{8}$ |
Given \\(a > 0\\), the function \\(f(x)= \frac {1}{3}x^{3}+ \frac {1-a}{2}x^{2}-ax-a\\).
\\((1)\\) Discuss the monotonicity of \\(f(x)\\);
\\((2)\\) When \\(a=1\\), let the function \\(g(t)\\) represent the difference between the maximum and minimum values of \\(f(x)\\) on the interval \\([t,t+3]\\). Find the minimum v... | \frac {4}{3} |
Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Calculate the number of ways to select exactly one person to serve for both days. | 60 |
Let \(ABC\) be a triangle such that \(\frac{|BC|}{|AB| - |BC|} = \frac{|AB| + |BC|}{|AC|}\). Determine the ratio \(\angle A : \angle C\). | 1 : 2 |
If the scores for innovation capability, innovation value, and innovation impact are $8$ points, $9$ points, and $7$ points, respectively, and the total score is calculated based on the ratio of $5:3:2$ for the three scores, calculate the total score of the company. | 8.1 |
A pyramid with a square base has all edges of 1 unit in length. What is the radius of the sphere that can be inscribed in the pyramid? | \frac{\sqrt{6} - \sqrt{2}}{4} |
The probability of an event occurring in each of 900 independent trials is 0.5. Find a positive number $\varepsilon$ such that with a probability of 0.77, the absolute deviation of the event frequency from its probability of 0.5 does not exceed $\varepsilon$. | 0.02 |
The distance between A and C is the absolute value of (k-7) plus the distance between B and C is the square root of ((k-4)^2 + (-1)^2). Find the value of k that minimizes the sum of these two distances. | \frac{11}{2} |
Given: $A=2a^{2}-5ab+3b$, $B=4a^{2}+6ab+8a$.
$(1)$ Simplify: $2A-B$;
$(2)$ If $a=-2$, $b=1$, find the value of $2A-B$;
$(3)$ If the value of the algebraic expression $2A-B$ is independent of $a$, find the value of $b$. | -\frac{1}{2} |
What is the smallest positive integer with exactly 10 positive integer divisors? | 48 |
A clerk can process 25 forms per hour. If 2400 forms must be processed in an 8-hour day, how many clerks must you hire for that day? | One clerk can process 25 x 8 = <<25*8=200>>200 forms in a day.
To process the forms for that day 2400/200 = <<2400/200=12>>12 clerks must be hired.
#### 12 |
Find the number of $x$-intercepts on the graph of $y = \sin \frac{1}{x}$ (evaluated in terms of radians) in the interval $(0.0001, 0.001).$ | 2865 |
Zig wrote four times as many books as Flo. If Zig wrote 60 books, how many books did they write altogether? | If Zig wrote 60 books, four times as many books as Flo, Flo wrote 60/4=<<60/4=15>>15 books
Together, they wrote 60+15=<<60+15=75>>75 books.
#### 75 |
A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$ , but $54210$ is not since ... | 168 |
Given an isosceles triangle $ABC$ with $AB = AC = 40$ units and $BC = 24$ units, let $CX$ be the angle bisector of $\angle BCA$. Find the ratio of the area of $\triangle BCX$ to the area of $\triangle ACX$. Provide your answer as a simplified fraction. | \frac{3}{5} |
Tonya has $150.00 on her credit card. If she leaves any balance on her card at the end of the month, she is charged 20% interest. If she makes a $50.00 payment on her card, what will be the new balance? | Her card has a $150.00 balance and she makes a $50.00 payment so the new balance is 150-50 = $<<150-50=100.00>>100.00
She didn't pay it off so she is charged 20% interest on her $100.00 balance so the interest is .20*100 = $20.00
Her balance was $100.00 and she was charged $20.00 in interest so her new balance is 100+2... |
Let $p$ and $q$ be constants. Suppose that the equation
\[\frac{(x+p)(x+q)(x-15)}{(x-5)^2} = 0\]
has exactly $3$ distinct roots, while the equation
\[\frac{(x-2p)(x-5)(x+10)}{(x+q)(x-15)} = 0\]
has exactly $2$ distinct roots. Compute $100p + q.$ | 240 |
The first and thirteenth terms of an arithmetic sequence are 5 and 29, respectively. What is the fiftieth term? | 103 |
Given circles ${C}_{1}:{x}^{2}+{y}^{2}=1$ and ${C}_{2}:(x-4)^{2}+(y-2)^{2}=1$, a moving point $M\left(a,b\right)$ is used to draw tangents $MA$ and $MB$ to circles $C_{1}$ and $C_{2}$ respectively, where $A$ and $B$ are the points of tangency. If $|MA|=|MB|$, calculate the minimum value of $\sqrt{(a-3)^{2}+(b+2)^{2}}$. | \frac{\sqrt{5}}{5} |
Let point $P$ lie on the curve $y= \frac {1}{2}e^{x}$, and point $Q$ lie on the curve $y=\ln (2x)$. Find the minimum value of $|PQ|$. | \sqrt {2}(1-\ln 2) |
How many integers $n$ satisfy $(n+3)(n-7) \le 0$? | 11 |
Stan drove 300 miles in 5 hours, 20 minutes. Next, he drove 360 miles in 6 hours, 40 minutes. What was Stan's average speed in miles per hour for the total trip? | 55 |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 23 |
Joseph has a refrigerator, a water heater in his house, and an electric oven that consumes power at different rates. The total amount of money that Joseph pays for the energy used by the refrigerator is three times the amount he pays for the power used by the water heater. If the electric oven uses power worth $500 in ... | If the electric oven uses power worth $500 in a month, twice what Joseph pays for the power the water heater uses, Joseph pays 1/2*$500=$<<500/2=250>>250 for power used by the water heater.
The total amount of money that Joseph pays for the power used by the water heater and the electric oven is $250+$500 = $<<250+500=... |
What is the largest $4$ digit integer congruent to $15 \pmod{22}?$ | 9981 |
Compute
\[\begin{vmatrix} 1 & \cos (a - b) & \cos a \\ \cos(a - b) & 1 & \cos b \\ \cos a & \cos b & 1 \end{vmatrix}.\] | 0 |
In quadrilateral ABCD, m∠B = m∠C = 120°, AB = 4, BC = 6, and CD = 7. Diagonal BD = 8. Calculate the area of ABCD. | 16.5\sqrt{3} |
The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side. | 13 |
John receives $100 from his uncle and gives his sister Jenna 1/4 of that money. He goes and buys groceries worth $40. How much money does John have remaining? | When John gives his sister 1/4 of his money, he parts with 1/4*100 = <<1/4*100=25>>25 dollars
Johns remains with 100-25 = <<100-25=75>>75 dollars.
When he buys groceries worth $40, he remains with 75-40 = <<75-40=35>>35 dollars
#### 35 |
If $f (x) = x^2 - 1$, what is the value of $f (-1)$? | 0 |
The sum of a negative integer $N$ and its square is 6. What is the value of $N$? | -3 |
In the Cartesian plane, let $A=(0,0), B=(200,100)$, and $C=(30,330)$. Compute the number of ordered pairs $(x, y)$ of integers so that $\left(x+\frac{1}{2}, y+\frac{1}{2}\right)$ is in the interior of triangle $A B C$. | 31480 |
What is the total number of digits used when the first 2500 positive even integers are written? | 9449 |
A line passes through the vectors $\mathbf{a}$ and $\mathbf{b}$. For a certain value of $k$, the vector
\[ k \mathbf{a} + \frac{5}{8} \mathbf{b} \]
must also lie on the line. Find $k$. | \frac{3}{8} |
Let squares of one kind have a side of \(a\) units, another kind have a side of \(b\) units, and the original square have a side of \(c\) units. Then the area of the original square is given by \(c^{2}=n a^{2}+n b^{2}\).
Numbers satisfying this equation can be obtained by multiplying the equality \(5^{2}=4^{2}+3^{2}\... | 15 |
The equation $z^6+z^3+1=0$ has complex roots with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$. | 160 |
The letters of the alphabet are each assigned a random integer value, and $H=10$. The value of a word comes from the sum of its letters' values. If $MATH$ is 35 points, $TEAM$ is 42 points and $MEET$ is 38 points, what is the value of $A$? | 21 |
For how many positive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$? | 212 |
In the parallelogram $ABCD$ , a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$ . If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$ , find the area of the quadrilateral $AFED$ . | 250 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\sin 2B= \sqrt{3}b\sin A$.
1. Find $B$;
2. If $\cos A= \dfrac{1}{3}$, find the value of $\sin C$. | \dfrac{2\sqrt{6}+1}{6} |
Given the function $f(x)={x^3}+\frac{{{{2023}^x}-1}}{{{{2023}^x}+1}}+5$, if real numbers $a$ and $b$ satisfy $f(2a^{2})+f(b^{2}-2)=10$, then the maximum value of $a\sqrt{1+{b^2}}$ is ______. | \frac{3\sqrt{2}}{4} |
What is the area, in square units, of a regular hexagon inscribed in a circle whose area is $324\pi$ square units? Express your answer in simplest radical form. | 486 \sqrt{3} |
Given a cylinder of fixed volume $V,$ the total surface area (including the two circular ends) is minimized for a radius of $R$ and height $H.$ Find $\frac{H}{R}.$ | 2 |
John runs a telethon to raise money. For the first 12 hours, he generates $5000 per hour. The remaining 14 hours, he generates 20% more per hour. How much total money does he make? | He gets 12*5000=$<<12*5000=60000>>60000 for the first 12 hours
For the next 14 hours, he gets 5000*.2=$<<5000*.2=1000>>1000 more per hour
So he gets 5000+1000=$<<5000+1000=6000>>6000 per hour
So he gets 6000*14=$<<6000*14=84000>>84000
So in total, he makes 84000+60000=$<<84000+60000=144000>>144,000
#### 144000 |
A farmer has twice as many pigs as cows, and 4 more cows than goats. If the farmer has 56 animals total, how many goats does he have? | Let x be the number of goats
Cows:4+x
Pigs:2(4+x)=8+2x
Total:x+4+x+8+2x=56
4x+12=56
4x=44
x=<<11=11>>11 goats
#### 11 |
A large puzzle costs $15. A small puzzle and a large puzzle together cost $23. How much would you pay for 1 large and 3 small puzzles? | A small puzzle costs $23 - $15 = $<<23-15=8>>8.
Three small puzzles cost $8 x 3 = $<<8*3=24>>24.
So, 1 large and 3 small puzzles cost $24 + $15 = $<<24+15=39>>39.
#### 39 |
Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$ . Find $\frac{100bdf}{ace}$ .
*Proposed by Andrew Wu* | 75 |
Bill and Joan both work for a library. 5 years ago, Joan had 3 times as much experience as Bill. Now she has twice as much experience as Bill. How many years of experience does Bill have now? | Let b be Bill's years of experience and j be Joan's years of experience. We know that j - 5 = 3(b - 5) and j = 2b.
Substituting the second equation into the first equation, we get 2b - 5 = 3(b - 5)
Multiplying through the parentheses, we get 2b - 5 = 3b - 15
Adding 15 to both sides, we get 2b + 10 = 3b
Subtracting 2b f... |
In the $xy$-plane, how many lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer pass through the point $(4,3)$? | 2 |
A room measures 16 feet by 12 feet and includes a column with a square base of 2 feet on each side. Find the area in square inches of the floor that remains uncovered by the column. | 27,072 |
Find the remainder when $x^{100}$ is divided by $(x + 1)^3.$ | 4950x^2 + 9800x + 4851 |
Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is *ocular* if it starts at $(0, 0)$ and passes through at least one point in $G$ . Let $A$ be the set of angle measures of acute angles formed by two distinc... | 1/722 |
Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room. | 360 |
Let $a$ and $b$ be angles such that $\cos a + \cos b = \frac{1}{2}$ and $\sin a + \sin b = \frac{3}{11}.$ Find
\[\tan \left( \frac{a + b}{2} \right).\] | \frac{6}{11} |
A local park has 70 pigeons that call the park home. Half of the pigeons are black, and 20 percent of the black pigeons are male. How many more black female pigeons are there than black male pigeons? | Black:70/2=<<70/2=35>>35 pigeons
Black Male: 35(.20)=7 pigeons
35-7=<<35-7=28>>28 female black pigeon
There are 28-7=<<28-7=21>>21 more black female pigeons
#### 21 |
In rectangle ABCD, AB=2, BC=3, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of EF. What is the area of the quadrilateral formed by the points A, E, H, and G? | 1.5 |
In a rectangular parallelepiped $ABCDEFGH$, the edge lengths are given as $AB = 30$, $AD = 32$, and $AA_1 = 20$. Point $E$ is marked at the midpoint of edge $A_1B_1$, and point $F$ is marked at the midpoint of edge $B_1C_1$. Find the distance between the lines $AE$ and $BF$. | 19.2 |
The area of the floor in a square room is 225 square feet. The homeowners plan to cover the floor with rows of 6-inch by 6-inch tiles. How many tiles will be in each row? | 30 |
Call a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number of ways to order the letters in $H M M T T H E M E T E A M$ to get an almost palindrome. | 2160 |
Ruiz receives a monthly salary of $500. If he received a 6% raise, how much will be Ruiz's new salary? | Ruiz has $500 x 6/100 = $<<500*6/100=30>>30 salary raise.
Thus, his new salary is $500 + $30 = $<<500+30=530>>530.
#### 530 |
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$ . | 13 |
$ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\overline{CD}$, then what is $\cos \angle AMB$? | \frac{1}{3} |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 15, what is the smallest value of $n$ such that $f(n) > 15$? | n = 255 |
What is the value of $n$ if $2^{n}=8^{20}$? | 60 |
From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows:
75, 80, 85, 65, 95, 100, 70, 55, 65, 75, 85, 110, 120, 80, 85, 80, 75, 90, 90, 95, 70, 60, 60, 75, 90, 95, 65, 75, 80, 80. The number of students in this ... | 630 |
To support the school outreach program, Einstein wants to raise $500 by selling snacks. One box of pizza sells for $12, a pack of potato fries sells for $0.30, and a can of soda at $2. Einstein sold 15 boxes of pizzas, 40 packs of potato fries, and 25 cans of soda. How much more money does Einstein need to raise to rea... | Einstein collected 15 x $12 = $<<15*12=180>>180 for selling pizzas.
He collected 40 x $0.30 = $<<40*0.30=12>>12 for the 40 packs of potato fries.
He collected 25 x $2 = $<<25*2=50>>50 for the 25 cans of soda.
Thus, the total amount section Einstein collected was $180 + $12 +$50 = $<<180+12+50=242>>242.
Therefore, Einst... |
A show debut and 200 people buy tickets. For the second showing three times as many people show up. If each ticket cost $25 how much did the show make? | The second show sold 200*3=<<200*3=600>>600 tickets
So there were a total of 200+600=<<200+600=800>>800 tickets sold
That means they made 800*25=$<<800*25=20000>>20,000
#### 20000 |
Lawrence worked 8 hours each day on Monday, Tuesday and Friday. He worked 5.5 hours on both Wednesday and Thursday. How many hours would Lawrence work each day if he worked the same number of hours each day? | 8 hours * 3 = <<8*3=24>>24 hours
5.5 * 2 = <<5.5*2=11>>11 hours
24 + 11 = <<24+11=35>>35 hours
35/7 = <<35/7=5>>5 hours
Lawrence would work 5 hours each of the 7 days in a week.
#### 5 |
The largest frog can grow to weigh 10 times as much as the smallest frog. The largest frog weighs 120 pounds. How much more does the largest frog weigh than the smallest frog? | To find the weight of the smallest frog, 120 pounds / 10 = <<120/10=12>>12 pounds for the smallest frog.
The difference is 120 - 12 = <<120-12=108>>108 pounds.
#### 108 |
The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 5 \\ 2 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{5}{2} \\ -1 \end{pmatrix}\]lie on a line. Enter the equation of this line in the form "$y = mx + b$". | y = -\frac{5}{2} x - \frac{29}{4} |
If an arc of $45^{\circ}$ on circle $A$ has the same length as an arc of $30^{\circ}$ on circle $B$, then what is the ratio of the area of circle $A$ to the area of circle $B$? Express your answer as a common fraction. | \frac{4}{9} |
What is the value of the least positive base ten number which requires six digits for its binary representation? | 32 |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 |
If Xiao Zhang's daily sleep time is uniformly distributed between 6 to 9 hours, what is the probability that his average sleep time over two consecutive days is at least 7 hours? | 7/9 |
In $\triangle ABC,$ $AB=AC=25$ and $BC=23.$ Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?
[asy]
real... | 50 |
The digits $0,1,2,3,4,5,6$ are randomly arranged in a sequence. What is the probability of obtaining a seven-digit number that is divisible by four? (The number cannot start with zero.) | 0.25 |
A rectangular room measures 12-feet by 6-feet. How many square yards of carpet are needed to cover the floor of the room? | 8 |
A circle with a radius of 2 passes through the midpoints of three sides of triangle \(ABC\), where the angles at vertices \(A\) and \(B\) are \(30^{\circ}\) and \(45^{\circ}\), respectively.
Find the height drawn from vertex \(A\). | 2 + 2\sqrt{3} |
A student travels from his university to his hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon. For the return trip, he drives his friend's truck, which averages only 15 miles per gallon. Additionally, they make a detour of 50 miles at an average of 10 miles per gallon in the truck. Calcul... | 16.67 |
Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the point of intersection of its diagonals, $O_1$ and $O_2$ are the centers of the inscribed circles of triangles $ABM$ and $CMD$ respectively, $K$ is the midpoint of the arc $AD$ that does not contain points $B$ and $C$, $\angle O_1 K O_2 = 60^{\circ}$, $K O_1 = 10... | 10 |
Tilly counts 120 stars to the east of her house and six times that number to the west. How many stars does she count total? | First find the total number of stars she saw in the west: 120 stars * 6 = <<120*6=720>>720 stars
Then add the number she saw in the east to find the total number: 720 stars + 120 stars = <<720+120=840>>840 stars
#### 840 |
Provided $x$ is a multiple of $27720$, determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+7)(3x+8)$ and $x$. | 168 |
Madison makes 30 paper boats and sets them afloat. 20% are eaten by fish and Madison shoots two of the others with flaming arrows. How many boats are left? | First find the number of boats eaten by fish: 30 boats * 20% = <<30*20*.01=6>>6 boats
Then subtract the two groups of boats that were destroyed from the original number to find the number left: 30 boats - 6 boats - 2 boats = <<30-6-2=22>>22 boats
#### 22 |
What is the largest digit $N$ for which $2345N$ is divisible by 6? | 4 |
Liu and Li, each with one child, go to the park together to play. After buying tickets, they line up to enter the park. For safety reasons, the first and last positions must be occupied by fathers, and the two children must stand together. The number of ways for these 6 people to line up is \_\_\_\_\_\_. | 24 |
How many solutions does the equation $\tan(2x)=\cos(\frac{x}{2})$ have on the interval $[0,2\pi]?$ | 5 |
A certain shopping mall sells a batch of brand-name shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to expand sales and reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that if the price of each shirt is r... | 20 |
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle{ABC}$? | $3\sqrt{3}$ |
Let $A B C D$ be a convex quadrilateral so that all of its sides and diagonals have integer lengths. Given that $\angle A B C=\angle A D C=90^{\circ}, A B=B D$, and $C D=41$, find the length of $B C$. | 580 |
Convert $10101_3$ to a base 10 integer. | 91 |
Given that the side lengths of triangle \( \triangle ABC \) are 6, \( x \), and \( 2x \), find the maximum value of its area \( S \). | 12 |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | 0 |
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