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CAT 2020 Slot 2 — QA Question 22

Number TheoryEasy

The number of pairs of integer (x, y) satisfying x ≥ y ≥ - 20 and 2x + 5y = 99 is

Answer & solution

Correct answer: 17

Answer: 17

Solution

Easy

Solve for xx in terms of yy; integrality forces yy to be odd. Then apply the constraints y≥−20y\ge -20 and x≥yx\ge y to bound yy, and count the odd values in the allowed range.

1

Express xx and force integrality.

2x+5y=99⇒x=99−5y2\begin{aligned} &2x + 5y = 99 \Rightarrow x = \frac{99-5y}{2} \end{aligned}

For xx to be an integer, 99−5y99-5y must be even, so 5y5y is odd, hence yy is odd.

2

Lower bound on yy. Given y≥−20y\ge -20 and yy odd, the smallest allowed yy is −19-19.

y≥−19(smallest odd ≥−20)\begin{aligned} &y \ge -19 \quad\text{(smallest odd } \ge -20\text{)} \end{aligned}
3

Upper bound from x≥yx\ge y. Substitute x=99−5y2x=\tfrac{99-5y}{2}.

99−5y2≥y⇒ 99−5y≥2y⇒ 99≥7y⇒y≤997≈14.14\begin{aligned} &\frac{99-5y}{2} \ge y\\ &\Rightarrow\ 99 - 5y \ge 2y\\ &\Rightarrow\ 99 \ge 7y \Rightarrow y \le \tfrac{99}{7} \approx 14.14 \end{aligned}

So the largest odd yy is 1313.

4

Count odd yy from −19-19 to 1313.

count=13−(−19)2+1=322+1=17\begin{aligned} &\text{count} = \frac{13-(-19)}{2} + 1 = \frac{32}{2}+1 = 17 \end{aligned}
Number of pairs=17\text{Number of pairs} = 17

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CAT 2020 Slot 2 QA Q22 — Number Theory solution | TheCATExam