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 = (673-k)/18 = 37 + (7-k)/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 2007-10-04 06:08:22