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Number Theory

Euler's Theorem

Intro

Concept Quizzes

Euler's Theorem Warmup
Fermat's Little Theorem
Euler's Totient Function
Euler's Theorem (NT)
Finding the last few digits of a power - Euler's Theorem
Diffie-Hellman
RSA

Challenge Quizzes

Euler's Theorem: Level 2 Challenges
Euler's Theorem: Level 3 Challenges
Euler's Theorem: Level 4 Challenges
Euler's Theorem: Level 5 Challenges

Euler's Totient Function

Given that $30=2\times3\times5$ , what is the value of the Euler totient function of 30, i.e. what is $\phi(30)$ ?

6 8 10 12

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by Brilliant Staff

Given that the only prime factors of a certain positive integer $n$ are 2 and 3, find (in terms of $n$ ) the number of positive integers less than or equal to $n$ that are coprime to $n$ .

\frac n2

\frac n3

\frac n6

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by Brilliant Staff

$\dfrac1{24} , \dfrac{2}{24}, \dfrac{3}{24}, \ldots , \dfrac{23}{24}, \dfrac{24}{24}$

How many of the fractions above cannot be reduced?

6 8 10 12

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by Brilliant Staff

$\dfrac1{32} , \dfrac{2}{32}, \dfrac{3}{32}, \ldots , \dfrac{31}{32}, \dfrac{32}{32}$

Which of the following is true regarding the 32 fractions above?

Statement 1: The number of fractions that cannot be reduced is less than the number of fractions that can be reduced.

Statement 2: The number of fractions that cannot be reduced is equal to the number of fractions that can be reduced.

Statement 3: The number of fractions that cannot be reduced is greater than the number of fractions that can be reduced.

Statement 1 Statement 2 Statement 3

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by Brilliant Staff

Let $A$ denote the number of positive integers less than or equal to 11 that are coprime with 11.

Let $B$ denote the number of positive integers less than or equal to 12 that are coprime with 12.

Which one is larger, $A$ or $B$ ?

A

is larger than

B

B

is larger than

A

They are equal

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by Brilliant Staff