The number of positive integer valued pairs (x, y) satisfying 4x - 17y = 1 and x ≤ 1000 is

Numbers Linear Equations

The number of positive integer valued pairs (x, y) satisfying 4x - 17y = 1 and x ≤ 1000 is

  1. 55
  2. 57
  3. 58
  4. 59

Answer

4x - 17y = 1

There are infinite solutions to this equation, however you have some constraints.

Positive integer value of x and y

x ≤ 1000

First solution is at y = 3, when x = 13

Second solution is at y = 7 when x = 30

You have to find how many terms are there in the sequence 13, 30, 47, ... (terms less than 1000)

All numbers are of the form 13 + 17a.

Divide 1000 by 17. You get 14 as remainder. So, 986 (= 17*58) is the greatest number less than 1000 divisible by 17. When you add 13 to it, you get 999. So 999 is the greatest number of this sequence.

In the first term in the sequence above, a = 0, in the last term, a = 58.

Hence number of such solutions = 59.

The correct option is D.