Is this fraction non-terminating?I recently stumbled upon an observation: the fraction x y terminates if and only if y only has prime factors 2 and 5For example: 1 20 = 1 2 ⋅ 2 ⋅ 5 = 0.05 1 6 = 1 2 ⋅ 3 = 0.1 6 ¯ I think this is true because fractions are in the form: a 10 + b 100 + c 1000 + … a 2 ⋅ 5 + b 2 ⋅ 2 ⋅ 5 ⋅ 5 + c 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5 ⋅ 5 + …How can I rigorously prove this?

Question

Is this fraction non-terminating?I recently stumbled upon an observation: the fraction       x    y   terminates if and only if   y only has prime factors   2 and   5For example:      1    20    =      1          2      ⋅      2      ⋅      5        =  0.05      1    6    =      1          2      ⋅      3        =  0.1            6      ¯      I think this is true because fractions are in the form:      a    10    +      b    100    +      c    1000    +  …      a          2      ⋅      5        +      b          2      ⋅      2      ⋅      5      ⋅      5        +      c          2      ⋅      2      ⋅      2      ⋅      5      ⋅      5      ⋅      5        +  …How can I rigorously prove this?

amanhantmk · Accepted Answer

HINT: You basically have it. To say that you have a terminating decimal means that      x    y    =      m          10      s      for some integer   m and positive integer   s. Since the only factors of       10    s   are ..., the only possible factors of   q are ... . (Why? What are you tacitly assuming about   x and   y?)

Is this fraction non-terminating? I recently stumbled upon an observation: the fraction x

Answered question

Answer & Explanation

New Questions in Pre-Algebra