Number of Integer Solutions with Inequality Constraint

CAT 2006 Slot 1 · QA · Easy · Quantitative Aptitude

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The number of solutions of the equation 2x+y=402x + y = 40 where both xx and yy are positive integers and x≤yx \le y is:

  1. A.

    7

  2. B.

    13

  3. C.

    14

  4. D.

    18

  5. E.

    20

Answer

13

Explanation

y=40−2xy = 40 - 2x. Since x,yx, y are positive integers, x≥1x \ge 1 and 40−2x≥1  ⟹  x≤1940 - 2x \ge 1 \implies x \le 19. We also require x≤y  ⟹  x≤40−2x  ⟹  3x≤40  ⟹  x≤13.33x \le y \implies x \le 40 - 2x \implies 3x \le 40 \implies x \le 13.33. So xx can take integer values from 1 to 13 inclusive. Total solutions =13= 13.

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