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Positive Integer System of Equations

XAT 2021 · Quantitative Ability & Data Interpretation · Hard · Algebra

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Find z, if it is known that: a: −y2+x2=20-y^2 + x^2 = 20 b: y3−2x2−4z≥−12y^3 - 2x^2 - 4z \ge -12 and c: x,yx, y and zz are all positive integers

  1. A.

    Any integer greater than 0 and less than 24

  2. B.

    24

  3. C.

    We need one more equation to find z

  4. D.

    6

  5. E.

    1

Answer

E

Explanation

From (a): (x−y)(x+y)=20(x-y)(x+y) = 20. Since x,yx, y positive integers, factors of 20 with same parity are (x+y=10,x−y=2)  ⟹  x=6,y=4(x+y=10, x-y=2) \implies x = 6, y = 4. Substitute into (b): 43−2(62)−4z≥−12  ⟹  64−72−4z≥−12  ⟹  −8−4z≥−12  ⟹  4z≤4  ⟹  z≤14^3 - 2(6^2) - 4z \ge -12 \implies 64 - 72 - 4z \ge -12 \implies -8 - 4z \ge -12 \implies 4z \le 4 \implies z \le 1. Since zz is a positive integer, z=1z = 1.

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