Ah, what you meant isn't that you get a constant, but that you get an integer.
Let's rearrange this:
Dmitry wrote:((3.14159265 - 3.1415926)/3.141592)^-1=62831840
First actually do that inversion from the start, giving this:
3.141592/(3.14159265 - 3.1415926)=62831840
Next multiply top/bottom of the fraction by the same power of ten to get rid of the decimal points:
314159200/(314159265 - 314159260)=62831840
Now rewrite it as the general case:
(Number formed by the first n-2 digits followed by 00) / (the nth digit) = an integer(?)
If the nth digit is 1, 2, 4, or 5, then it will automatically work there because any number ending in 00 is a multiple of 100, and so will also be a multiple of 1, 2, 4, or 5.
If the nth digit is 3, 6, 7, 8, or 9 then it will not always work, and it will generally just be coincidence if it does. With pi, it is just coincidence that 3141 is divisible by 9. However at that 6 you get
31415900/6 = 5235983.3333333333333333333333333
and at the next 3 you get
3141592600/3 = 1047197533.3333333333333333333333
so it doesn't always work for pi.
[Edited for typo]