We should use base 12!

Number Theory Level 2

Which of the following is an infinite repeating decimal base 12 ?

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1 2 \dfrac{1}{2} 1 5 \dfrac{1}{5} 1 4 \dfrac{1}{4} 1 3 \dfrac{1}{3}

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Alex G by Alex G , 22, USA

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1 solution

Alex G
May 4, 2016

A fraction will be an infinite repeating decimal in base n n if the denominator does not divide n n . Hence, 1 5 \dfrac{1}{5} is an infinite repeating decimal in base 12 12 .

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Correction: A fraction will be a recurring decimal in base n n if the denominator has a prime factor which doesn't appear in the prime factorization of n n . e.g. ( 1 1 2 2 ) 10 = ( 1 144 ) 10 = ( 1 1 0 2 ) 12 = ( 1 100 ) 12 = ( 0.01 ) 12 \large{(\frac{1}{12^2})_{10} = (\frac{1}{144})_{10} = (\frac{1}{10^2})_{12} = (\frac{1}{100})_{12} = (0.01)_{12}} (Subscript (which is in base 10) is indicating the base.)

Jesse Nieminen - 5 years, 1 month ago

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