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 a−b) n=9ab.
Case 1: (9,n)=1: 9 | 10a−b ⇒ 9 | a−b ⇒a=b ⇒ 9an=9a2 ⇒ n=a=b (trivial)
Case 2: (9,n)=9: 10a−b=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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