elementary number theory - Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$

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 $\rm\: (10\ a + n)/(10\ n + b) = a/b\:.\:$ Thus $\rm\ (10\ a-b)\ n = 9\:a\:b\:.$

Case 1: $\rm\:(9,n) = 1\::\ \: 9\ |\ 10\:a-b\ \Rightarrow\ 9\ |\ a-b\ \Rightarrow a=b\ \Rightarrow\ 9\:a\:n = 9\:a^2\ \Rightarrow\ n=a=b\:$ (trivial)

Case 2: $\rm\:(9,n) = 9\::\ 10\:a-b = a\:b\ \Rightarrow\ a|b,\ 10 = (b/a)\:(a+1)\:$ so $\rm\ a,b = 1,5\:$ or $\: 4,8\:$

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

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