Determine all possible positive real p satisfying 18[p] + 35{p} = 673, where [y] denotes the greatest integer ≤ y and {y} = y  [y]
Since 673 is an integer and so is 18[p] then 35{p} is an integer, so {p} (which is in the range [0,1]) = k/35 where k is an integer in [0,34]. Then p = n + k/35 where n = [p]. The equation reduces to:
18n + k = 673
Write n in terms of k:
n = (673k)/18 = 37 + (7k)/18
n is an integer so the only values of k that have solutions are k=7 mod 18 and k in [0,34] which gives either k=7 or k=25.
When k = 7, n=37 and p = 37 7/35
when k= 25, n=36 and p = 36 25/35
Thanks to the two restrictions on k (nonnegative integer less than 35 and =7 mod 18) we can be certain there are no other solutions.
These are the same results as Chesca Ciprian's solution, who therefore deserves all credit.

Posted by Paul
on 20071004 06:08:22 