All Questions ⁺⁰236088 Questions

0

14

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+622

Number Theory

A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. How many prime numbers are divisors of $n?$

cooIcooIcooI17 Jul 7, 2024

0

15

1

+622

Number Theory

A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. What is the smallest possible value of $n?$

cooIcooIcooI17 Jul 7, 2024

Jul 6, 2024

-2

15

1

+330

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 $\phi(1200)$?

Stanry Jul 6, 2024

-2

10

1

+330

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 $\phi(191)$?

Stanry Jul 6, 2024

-2

11

2

+330

Number Theory

The number $100$ has four perfect square divisors, namely $1,$ $4,$ $25,$ and $100.$
What is the smallest positive integer that has exactly $2$ perfect square divisors?

Stanry Jul 6, 2024

0

13

2

+23

Binomial theorem problem

Find the largest prime factor of 1,005,010,010,005,001 without using brute force.

Coolmonkey Jul 6, 2024

0

22

0

+842

Number Theory

Find the last two digits of $16*{37}$.
(Compute the remainder when $16*{37}$ is divided by 100.)

RedDragonl Jul 6, 2024

0

13

0

+842

Number Theory

Let $a$ be an integer such that $a \equiv 5 \pmod{7}$. Find the value of $a + 1 \pmod{7}$. Express your answer as a residue between $0$ and the modulus.

RedDragonl Jul 6, 2024

0

11

0

+842

Number Theory

Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$. If $a \neq 0,$ then find the value of $a$.

RedDragonl Jul 6, 2024

0

21

0

+842

Number Theory

Let $a$ be an integer such that $0 \le a \le 7$ and $a^2 \equiv a \pmod{8}$. If $a \neq 0,$ then find the value of $a$.

RedDragonl Jul 6, 2024

0

13

0

+976

Number Theory

Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

Hi6942O Jul 6, 2024

0

17

1

+976

Number Theory

The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
10011010011ABC110_2.
Compute the ordered triple of digits $(A,B,C)$.

Hi6942O Jul 6, 2024

0

14

1

+976

Number Theory

The Fibonacci sequence is defined by F_1 = F_2 = 1 and F_{n + 2} = F_{n + 1} + F_n. Find the remainder when F_{1000} is divided by 17.

Hi6942O Jul 6, 2024

0

20

1

+976

Number Theory

Fill in the coefficients:
n(n - 1)(n + 17)(n - 21)(n - 33) = n^5 + ___ n^4 + ___ n^3 + ___ n^2 + ___n + ___

Hi6942O Jul 6, 2024

0

10

2

+8

Precalculus

Calculate

As usual, the output of an inverse trig function should be in radians.

abc12345678 Jul 6, 2024

0

12

1

+948

Number Theory

For an integer $n,$ let $f(n)$ be the remainder when $n^8 + n^{16}$ is divided by $5.$ Compute $f(0) + f(1) + f(2) + f(3) + f(4).$

Rangcr897 Jul 6, 2024

0

21

1

+948

Number Theory

For a certain positive integer $n$, the number $n^{6873}$ leaves a remainder of $3$ when divided by $23.$ What remainder does $n$ leave when divided by $23$?

Rangcr897 Jul 6, 2024

0

17

0

+948

Number Theory

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

Rangcr897 Jul 6, 2024

0

17

0

+948

Number Theory

Given
a &\equiv 1 \pmod{7} \\
b &\equiv 2 \pmod{7} \\
c &\equiv 6 \pmod{7} read more ..

Rangcr897 Jul 6, 2024

0

17

0

+948

Counting

Starting with the A and moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?
A
RR
CCC read more ..

Rangcr897 Jul 6, 2024

0

14

0

+948

Probability

Alex chooses a number at random from the set $\{1, 2, 3, \dots, 5\}.$ Winnie also chooses a number at random from the same set. (They can choose the same number.) What is the probability that the product of their numbers is even?

Rangcr897 Jul 6, 2024

0

13

1

+948

Probability

The following cards are split into three piles at random, so that every pile contains the same number of cards. What is the probability that every pile contains an Ace?
Ace of spades
Ace of hearts
Ace of diamonds read more ..

Rangcr897 Jul 6, 2024

0

20

0

+948

Number Theory

How many lattice points (points with integer coordinates) are on the line segment whose endpoints are $(3,17)$ and $(82,150)?$ (Include both endpoints in your count.)

Rangcr897 Jul 6, 2024

0

16

0

+948

Number Theory

On planet Enigma, the residents use a currency called the confusion. There are only two confusion bills on Enigma, one worth $7$ confusions and the other worth $12$ confusions. There are also some coins of smaller value, but each weighs over read more ..

Rangcr897 Jul 6, 2024

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