Determine all possible pairs (p, q) of positive integers with p < q < 400 where gcd(p,q)=1 such that the first four digits following the decimal point in the base ten expansion of p/q is 2011.

*** For an extra challenge, solve this puzzle without using a computer program.

.2011 is just over 1/5 so we can do a quick search for numbers like 1/5 but with the denominator slightly smaller

Finding the limits:
x/(5x-1) = .2011
x = 36.56
x/(5x-1) = .2012
x = 33.53
So x = 34, 35, 36
and we have the fractions
34/169
35/174
36/179

Making the demoninator (5x-2) instead gives limits of 67.06 and 73.12
so we have the numerators 68, 69, 70, 71, 72, 73
and the fractions
68/338
69/343
70/348
71/353
72/358
73/363

Making the denominator (5x-3) gives the numerators
101 through 109 but the denominators are all over 400
so the 9 solutions above are all of them.

Posted by Jer on 2011-11-22 11:05:43