Fraction TransformsHere's a number theory problem I'm having some difficulty with:Say we transform a fraction by the following rule: we start with some fraction m n with m &gt; n and then convert it to f r n , where f = ⌊ m / n ⌋ and r = m mod n . We repeat the process until our fraction is less than 1 or an integer. Given an n is there an m so that we end up with a fraction less than 1?For n = 2 it isn't that hard, 3, 7, 15, 31, etc. work, but what about other values of n?

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Fraction TransformsHere&#039;s a number theory problem I&#039;m having some difficulty with:Say we transform a fraction by the following rule: we start with some fraction       m    n    with    m  &amp;gt;  n and then convert it to             f      r        n  , where   f  =  ⌊  m      /    n  ⌋ and   r  =  m  mod      n  . We repeat the process until our fraction is less than 1 or an integer. Given an n is there an m so that we end up with a fraction less than 1?For   n  =  2 it isn&#039;t that hard, 3, 7, 15, 31, etc. work, but what about other values of   n?

Belen Bentley · Accepted Answer

One can merely work backwards. Suppose we want to end up at    1      /    n. Then the preceding term could be   1  +      1    n    =  (  n  +  1  )      /    n. Then the preceding term could be   (  n  +  1  )  +      1    n    =  (      n    2    +  n  +  1  )      /    n. By inspection, any improper fraction of the form   (      n    k    +      n          k      −      1        +  ⋯  +      n    2    +  n  +  1  )      /    n will end up at   1      /    n

Fraction Transforms Here's a number theory problem I'm having some difficulty with: Say we transfo

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