All Questions ⁺⁰236027 Questions

0

14

1

+2653

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.

LiIIiam0216 Jul 24, 2024

0

12

2

+355

modulo help

(a) Find a number that solves the congruence .
(b) Find a number that solves the congruence .

Lilliam0216 Jul 24, 2024

0

16

0

+510

Number Theory

What are the first 5 digits after the decimal point (technically the hexadecimal point...) when the fraction \frac{2}{23} is written in base 16?
Express your answer as a five digit hexadecimal number. You do not need to include read more ..

onyuIee Jul 24, 2024

0

16

1

+510

Number Theory

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

onyuIee Jul 24, 2024

0

11

1

+510

Number Theory

Find the six-digit number abcdef such that
0.\overline{abcdef} = \frac{2}{17},
without a calculator.

onyuIee Jul 24, 2024

0

10

1

+510

Number Theory

Find the greatest prime divisor of the value of the arithmetic series
1 + 2 + 3 + \dots + 135 + 136 + 137 + 138 + 139 + 140.

onyuIee Jul 24, 2024

0

21

1

+510

Number Theory

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

onyuIee Jul 24, 2024

0

15

1

+510

Number Theory

What is the smallest prime divisor of 5^{19} * 7^{13} * 3^{31}?

onyuIee Jul 24, 2024

0

8

1

+510

Number Theory

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

onyuIee Jul 24, 2024

0

12

1

+363

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence. If
\[a_n = a_{n - 1} + a_{n - 2}\]
for all $n \ge 3,$ and $a_{11} = 1$ and $a_{10} = 4,$ then find $a_6.$

MeIdHunter Jul 24, 2024

0

9

1

+363

Algebra

Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

MeIdHunter Jul 24, 2024

0

9

1

+363

Algebra

Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.

MeIdHunter Jul 24, 2024

0

8

1

+893

Algebra

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

AnswerscorrectIy Jul 24, 2024

0

7

1

+893

Algebra

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

AnswerscorrectIy Jul 24, 2024

0

19

0

+893

Algebra

Let $f(x) = 3x - 8 + 4x^2$ and $g(x) = 15x + c - 3x^2$. Find $c$ if $(f \circ g)(x) = (g \circ f)(x)$ for all $x$.

AnswerscorrectIy Jul 24, 2024

0

18

0

+363

Algebra

Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be an arithmetic sequence. Let $S_n$ denote the sum of the first $n$ terms. If $S_{20} = \frac{1}{5}$ and $S_{10} = 0,$ then find $S_{70}.$

MeIdHunter Jul 24, 2024

0

20

1

+363

Algebra

Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 20, Shirley runs screaming from the room. What number does Laverne say that sends read more ..

MeIdHunter Jul 24, 2024

0

11

1

+363

Algebra

The equation of the line below can be expressed in the form
\frac{x}{a} + \frac{y}{b} = 1.
Find a.
The lines passes through (5,8) and (-4,17).

MeIdHunter Jul 24, 2024

0

18

1

+363

Algebra

Evaluate the infinite geometric series
0.4 + 0.036 + 0.0000324 + ...
Express your answer as a fraction with integer numerator and denominator.

MeIdHunter Jul 24, 2024

0

10

1

+363

Algebra

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator read more ..

MeIdHunter Jul 24, 2024

0

11

1

+1839

Algebra

Find the number of ordered pairs $(a,b)$ of integers such that
\frac{a + 2}{a + 1} = \frac{b}{3}.

ABJeIIy Jul 24, 2024

0

23

0

+1839

Algebra

The system of equations
\frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 1, \quad \frac{yz}{y + z} = 2
has exactly one solution. What is $z$ in this solution?

ABJeIIy Jul 24, 2024

0

10

1

+363

Algebra

Let a_1, a_2, a_3, \dots, a_{10}, a_{11}, a_{12} be an arithmetic sequence. If $a_1 + a_3 + a_5 + a_7 + a_9 + a_{11} = 0$ and $a_2 + a_4 + a_6 + a_8 + a_{10} + a_{12} = 0$, then find $a_1$.

MeIdHunter Jul 24, 2024

0

10

3

+363

Algebra

When the same constant is added to the numbers 60, 120, and 160, a three-term geometric sequence arises. What is the common ratio of the resulting sequence?

MeIdHunter Jul 24, 2024

0

19

2

+1839

Algebra

In a store window, there was a box of berries having a total weight of 200 kg. The berries were 99% water, by weight. After two days in the sun, the water content of the berries was only 90%, by weight. What was the total weight of read more ..

ABJeIIy Jul 24, 2024

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