Number Theory
# Euler's Theorem

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

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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