Diophantine Equation: Positive Integer Pairs

CAT 1999 Slot 1 · Quantitative Ability · Medium · Number Systems

This is a medium Quantitative Ability question from the CAT 1999 Slot 1 paper. It tests Number Systems. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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

A.
59
B.
57
C.
55
D.
58

Answer

Explanation

Since $4x - 17y = 1$ , $17y = 4x - 1 \equiv -1 \equiv 3 \pmod 4$ . Since $17 \equiv 1 \pmod 4$ , $y \equiv 3 \pmod 4$ . So $y$ can be $3, 7, 11, \dots$ . For $x \le 1000$ , $4x \le 4000 \implies 17y + 1 \le 4000 \implies 17y \le 3999 \implies y \le 235.23$ . The terms for $y$ form an AP: $3, 7, 11, \dots, 235$ . $235 = 3 + (n-1)4 \implies 232 = 4(n-1) \implies n - 1 = 58 \implies n = 59$ .

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Diophantine Equation: Positive Integer Pairs

Explanation

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