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# Euler's Theorem

Euler's theorem relate to the remainder of various powers and has applications ranging from modern cryptography to recreational problem-solving.

\[ \dfrac1{24} , \dfrac{2}{24}, \dfrac{3}{24}, \ldots , \dfrac{23}{24}, \dfrac{24}{24} \]

How many of the fractions above **cannot** be reduced?

\[ \dfrac1{32} , \dfrac{2}{32}, \dfrac{3}{32}, \ldots , \dfrac{31}{32}, \dfrac{32}{32} \]

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.

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 \(B\)?

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