All Questions ⁺⁰236027 Questions

0

18

0

+2653

Number Theory

Find the remainder when $100^{100}$ is divided by $18$.

LiIIiam0216 Jul 25, 2024

0

14

0

+2653

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

25

0

+2653

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

18

0

+2653

Number Theory

Find the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by $2$.

LiIIiam0216 Jul 25, 2024

+1

11

1

+510

Algebra

Find all the solutions to
\frac{x+4}{x+5} = \frac{x-3}{2} + \frac{x + 7}{5}

onyuIee Jul 25, 2024

0

12

2

+510

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?

onyuIee Jul 25, 2024

0

15

1

+510

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

13

2

+510

Algebra

Find the sum of the squares of the roots of $2x^2+4x-1=x^2-8x+3$.

onyuIee Jul 25, 2024

0

20

1

+510

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 ●

onyuIee Jul 25, 2024

0

15

1

+510

Algebra

If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?

NotThatSmart ●

onyuIee Jul 25, 2024

Jul 24, 2024

0

14

1

+510

Algebra

Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6,$ then what is $a^3 + b^3?$

onyuIee Jul 24, 2024

0

11

1

+893

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 ●

AnswerscorrectIy Jul 24, 2024

0

11

1

+893

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.

AnswerscorrectIy Jul 24, 2024

0

14

0

+893

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

6

2

+363

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 $ABC.$

MeIdHunter Jul 24, 2024

0

16

0

+363

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

8

1

+363

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 $BX$?

MeIdHunter Jul 24, 2024

0

17

1

+363

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 $ABD$.

MeIdHunter Jul 24, 2024

0

13

1

+2653

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 $\phi(2835)$?

LiIIiam0216 Jul 24, 2024

0

11

0

+2653

Number Theory

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

LiIIiam0216 Jul 24, 2024

0

19

1

+2653

Number Theory

Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$

LiIIiam0216 Jul 24, 2024

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