Sunday, February 17, 2019

elementary number theory - Bad Fraction Reduction That Actually Works



1664=1/6/64=14




This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there are other such examples. Is there a systematic way to generate examples of this kind of bad fraction reduction?


Answer



It's easy to find them all. Suppose (10 a+n)/(10 n+b)=a/b. Thus  (10 ab) n=9ab.



Case 1: (9,n)=1: 9 | 10ab  9 | ab a=b  9an=9a2  n=a=b (trivial)



Case 2: (9,n)=9: 10ab=ab  a|b, 10=(b/a)(a+1) so  a,b=1,5 or 4,8



which yields the solutions:  19/95=1/5, and 49/98=1/2.  Similar analysis of the remaining




Case 3: (9,n)=3: yields 16/64=1/4, and 26/65=2/5.


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