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Post New Question
All Questions
+0
234503 Questions
0
5
0
+2633
Number Theory
Find the remainder when $100^{100}$ is divided by $18$.
LiIIiam0216
Jul 25, 2024
0
4
0
+2633
Number Theory
(a) Find the last two digits of $9*5*3$.
(b) Find the last two digits of $1^{2^3}$. (By convention, exponent towers are evaluated from the top down, so $1^{2^3} = 1^{(2^3)}$.)
LiIIiam0216
Jul 25, 2024
0
7
0
+2633
Number Theory
Given
a &\equiv 1 \pmod{7} \\
b &\equiv 2 \pmod{7} \\
c &\equiv 6 \pmod{7}
read more ..
LiIIiam0216
Jul 25, 2024
0
3
0
+2633
Number Theory
Find the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by $2$.
LiIIiam0216
Jul 25, 2024
+1
3
1
+299
Algebra
Find all the solutions to
\frac{x+4}{x+5} = \frac{x-3}{2} + \frac{x + 7}{5}
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onyuIee
Jul 25, 2024
0
2
2
+299
Algebra
The solutions to
2x^2 - 10x + 13 = -6x^2 - 18x - 15
are a+bi and a-bi, where a and b are positive. What is a\cdot b?
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onyuIee
Jul 25, 2024
0
3
1
+299
Algebra
The quadratic equation $x^2-mx+24 = 10$ has roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible?
NotThatSmart
●
onyuIee
Jul 25, 2024
0
2
2
+299
Algebra
Find the sum of the squares of the roots of $2x^2+4x-1=x^2-8x+3$.
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onyuIee
Jul 25, 2024
0
7
1
+299
Algebra
Let $m$ be a real number. If the quadratic equation $x^2+mx+4 = 2x^2 + 17x + 8$ has two distinct real roots, then what are the possible values of $m$? Express your answer in interval notation.
NotThatSmart
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onyuIee
Jul 25, 2024
0
2
1
+299
Algebra
If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?
NotThatSmart
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onyuIee
Jul 25, 2024
Jul 24, 2024
0
4
1
+299
Algebra
Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6,$ then what is $a^3 + b^3?$
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onyuIee
Jul 24, 2024
0
4
1
+593
Geometry
A line and a circle intersect at $A$ and $B,$ as shown below. Find the distance between $A$ and $B$.
The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.
NotThatSmart
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AnswerscorrectIy
Jul 24, 2024
0
1
1
+593
Geometry
Let $k$ be a positive real number. The line $x + y = 3 + k$ and the circle $x^2 + y^2 = k$ are drawn. Find $k$ so that the line is tangent to the circle.
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AnswerscorrectIy
Jul 24, 2024
0
3
0
+593
Geometry
Two circles intersect at two points, $P$ and $Q$. The equations of the two circles are $x^2 + (y - 1)^2 = 1$ and $(x - 1)^2 + y^2 = 1$. Find the length PQ.
AnswerscorrectIy
Jul 24, 2024
0
2
2
+126
Geometry
In triangle $ABC,$ the angle bisector of $\angle BAC$ meets $\overline{BC}$ at $D.$ If $\angle BAC = 60^\circ,$ $\angle CAD = 45^\circ,$ and $AD = 24,$ then find the area of triangle $ABC.$
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MeIdHunter
Jul 24, 2024
0
5
0
+126
Geometry
In triangle $ABC,$ $AB = 15,$ $BC = 9,$ and $AC = 10.$ Find the length of the shortest altitude in this triangle.
MeIdHunter
Jul 24, 2024
0
2
1
+126
Geometry
In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?
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MeIdHunter
Jul 24, 2024
0
4
1
+126
Geometry
In triangle $ABC$, point $D$ is on side $\overline{AC}$ such that line segment $\overline{BD}$ bisects $\angle ABC$. If $\angle A = 45^\circ$, $\angle C = 45^\circ$, and $AC = 12$, then find the area of triangle $ABD$.
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MeIdHunter
Jul 24, 2024
0
3
1
+2633
Number Theory
For a positive integer n, \phi(n) denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$. What is $\phi(2835)$?
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LiIIiam0216
Jul 24, 2024
0
6
0
+2633
Number Theory
How many numbers between $1000$ and $2000$ leave a remainder of $3$ when divided by $47?$
LiIIiam0216
Jul 24, 2024
0
3
1
+2633
Number Theory
Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$
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LiIIiam0216
Jul 24, 2024
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