We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
184
2
avatar+322 

For how many integers between 1 and 11 (inclusive) is  n/12 a repeating decimal?

I can obviosly guess and check this, but I want the actual way. 

 Feb 22, 2019
 #1
avatar+5782 
+1

If a rational number's decimal representation doesn't terminate then it repeats.

 

A rational numbers decimal representation will terminate only if the denominator's only prime factors are 2 and 5.

 

With 12 in the denominator we need a 3 in the numerator to ensure the decimal terminates.

 

Thus 3/12, 6/12, 9/12 will all terminate

 

1/12, 2/12, 4/12, 5/12, 7/12, 8/12, 10/12, 11/12 will all repeat

 Feb 22, 2019
 #2
avatar+23064 
+3

 For how many integers n between 1 and 11 (inclusive) n/12 is a repeating decimal?

 

\(\begin{array}{|r|r|r|r|r|} \hline & & & & & \text{prime number only } \\ & & & \text{factorize} & \text{prime numbers} & 2 \text{ or } 5 \\ n & \dfrac{n}{12} & \text{cancel} & \text{the denominator} & \text{in the denominator}& \text{will terminate} \\ \hline 1 & \dfrac{1}{12} & \dfrac{1}{12} & \dfrac{1}{2^2\cdot 3} & 2,\ 3 \\ \hline 2 & \dfrac{2}{12} & \dfrac{1}{6} & \dfrac{1}{2 \cdot 3} & 2,\ 3 \\ \hline 3 & \dfrac{3}{12} & \dfrac{1}{4} & \dfrac{1}{2^2} & 2 & \checkmark \Rightarrow \text{terminate} \\ \hline 4 & \dfrac{4}{12} & \dfrac{1}{3} & \dfrac{1}{3} & 3 \\ \hline 5 & \dfrac{5}{12} & \dfrac{5}{12} & \dfrac{1}{2^2\cdot 3} & 2,\ 3 \\ \hline 6 & \dfrac{6}{12} & \dfrac{1}{2} & \dfrac{1}{2} & 2 & \checkmark \Rightarrow \text{terminate} \\ \hline 7 & \dfrac{7}{12} & \dfrac{7}{12} & \dfrac{7}{2^2\cdot 3} & 2,\ 3 \\ \hline 8 & \dfrac{8}{12} & \dfrac{2}{3} & \dfrac{2}{3} & 3 \\ \hline 9 & \dfrac{9}{12} & \dfrac{3}{4} & \dfrac{3}{2^2} & 2 & \checkmark \Rightarrow \text{terminate} \\ \hline 10 & \dfrac{10}{12} & \dfrac{5}{6} & \dfrac{5}{2\cdot 3} & 2,\ 3 \\ \hline 11 & \dfrac{11}{12} & \dfrac{11}{12} & \dfrac{11}{2^2\cdot 3} & 2,\ 3 \\ \hline \end{array} \)

 

see: https://gmatclub.com/forum/math-number-theory-88376.html

 

laugh

 Feb 22, 2019

7 Online Users