All Questions ⁺⁰237313 Questions

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24

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Number Theory

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

LiIIiam0216 Jul 15, 2024

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24

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+2729

Number Theory

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

LiIIiam0216 Jul 15, 2024

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35

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+1208

Number Theory

A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?

AnswerscorrectIy Jul 15, 2024

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24

1

+1208

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?

AnswerscorrectIy Jul 15, 2024

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31

1

+1208

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 $\phi(191)$?

AnswerscorrectIy Jul 15, 2024

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22

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+1208

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 $\phi(1200)$?

AnswerscorrectIy Jul 15, 2024

0

20

0

+2729

Number Theory

Which of the residues $0,$ $1,$ $2,$ $\dots,$ $11$ satisfy the congruence $3x \equiv 8 \pmod{5}?$
Give your answer as a list, separated by commas, in order from least to greatest.

LiIIiam0216 Jul 15, 2024

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29

0

+2729

Number Theory

Which of the residues $0,$ $1,$ $2,$ $3,$ $4$ satisfy the congruence $3x \equiv 0 \pmod{5}?$
Give your answer as a list, separated by commas, in order from least to greatest.

LiIIiam0216 Jul 15, 2024

0

28

0

+2729

Number Theory

Which of the residues $0,$ $1,$ $2,$ $3,$ $4,$ $5$ satisfy the congruence $x^2 \equiv 0 \pmod{6}?$
Give your answer as a list, separated by commas, in order from least to greatest.

LiIIiam0216 Jul 15, 2024

0

35

1

+2729

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.

LiIIiam0216 Jul 15, 2024

0

39

0

+2729

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 $a$.

LiIIiam0216 Jul 15, 2024

0

40

0

+2729

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 $a$.

LiIIiam0216 Jul 15, 2024

0

13

1

+1208

Number Theory

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

AnswerscorrectIy Jul 15, 2024

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36

1

+1208

Number Theory

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

AnswerscorrectIy Jul 15, 2024

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23

1

+1208

Number Theory

Calculate $40*9 \pmod{23}.$ Express your answer as a non-negative integer that is less than $23$.

AnswerscorrectIy Jul 15, 2024

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26

1

+1208

Number Theory

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

AnswerscorrectIy Jul 15, 2024

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29

2

+1208

Number Theory

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

AnswerscorrectIy Jul 15, 2024

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41

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+1208

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 $1^{2^3} = 1^{(2^3)}$.)

AnswerscorrectIy Jul 15, 2024

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41

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+157

Pls help

Mr Ravi spent 1/4 ofhis money on a shirt and 5/6 of his remaining money on a badminton racket, a pair of shorts and 2 pairs of shoes. The badminton racket cost 4 times as much as a pair of shoes while the pair of shorts cost 4 times read more ..

sugabreaddoughman Jul 15, 2024

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33

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+157

Help

Watermelons and mangoes are sold at a fruit stall. There were 13 more watermelons than mangoes at first. After 3/5 of the watermelons and 5/9 of the mangoes were sold, there was an equal number of each type of fruit left unsold.
a) read more ..

sugabreaddoughman Jul 15, 2024

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37

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+157

Need guide

Baker Yeo sold 120 buns in the morning and 40% of the remaining buns in the afternoon. The number of buns left was 1/3 of the number of buns he had baked. How many buns did he sell in the afternoon?

sugabreaddoughman Jul 15, 2024

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35

1

+157

Help

In a fruit stall, there were 119 more apples than oranges. 6/7 of the apples and 4/5 of the oranges were sold. As a result, there were 33 more oranges than apples left.

(a) How many oranges were there at first?
(b) read more ..

sugabreaddoughman Jul 15, 2024

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30

1

+1208

Number Theory

(a) Find the last two digits of $9*7*8$.
(b) Find the last two digits of $2^{3^4}$. (By convention, exponent towers are evaluated from the top down, so $2^{3^4} = 2^{(3^4)}$.)

AnswerscorrectIy Jul 15, 2024

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20

1

+1208

Number Theory

(a) Find the last two digits of $9*7*11$.
(b) Find the last two digits of $4^{5^6}$. (By convention, exponent towers are evaluated from the top down, so $4^{5^6} = 4^{(5^6)}$.)

AnswerscorrectIy Jul 15, 2024

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