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

doodverft05

doodverft05

Answered question

2022-06-23

Fraction Transforms
Here'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 > 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?

Answer & Explanation

Belen Bentley

Belen Bentley

Beginner2022-06-24Added 28 answers

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

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