CAT 2017 Set-2 | Question: 94

578 views

How many different pairs $(a,b)$ of positive integers are there such that $a\leq b$ and $1/a+1/b=1/9$

1. None of these
2. $2$
3. $0$
4. $1$

Given that, $\dfrac{1}{a}+\dfrac{1}{b} = \dfrac{1}{9}\;; \;a,b\in \mathbb{Z}^{+}$ and $a\leq b$

$\Rightarrow \dfrac{a+b}{ab} = \dfrac{1}{9}$

$\Rightarrow 9a+9b = ab$

$\Rightarrow 9a+9b-ab = 0$

$\Rightarrow ab-9a-9b = 0$

$\Rightarrow ab-9a-9b+81 = 81$

$\Rightarrow (a-9)(b-9) = 81$

We can factorize $81$ such that $a-9\leq b-9 \Rightarrow \boxed{a\leq b}$

$\qquad \qquad \begin{array} {ccc} \underline{a-9}& \leq & \underline{b-9} \\ 1 & & 81 \\ 3 & & 27 \\ 9 & & 9\end{array}$

$\therefore$ Only three pairs are possible.

Correct Answer $:\text{A}$

Related questions

780
views
780 views
If $9^{\left ( x-1/2 \right )}-2^{\left ( 2x-2 \right )}=4^{x}-3^{\left (2x-3 \right )}$, then $x$ is$3/2$$2/5$$3/4$$4/9 501 views 1 answers 1 votes 501 views If a, b, c and d are integers such that a+b+c+d=30 , then the minimum possible value of ( a-b )^{2}+( a-c )^{2}+( a-d)^{2} is 1$$2$$5$$6$
486
views
The numbers $1, 2,\dots$,$9$ are arranged in a $3 \times 3$ square grid in such a way that each number occurs once and the entries along each column, each row, and each o...
In a $10$ km race, $\text{A, B}$ and $\text{C}$, each running at uniform speed, get the gold, silver, and bronze medals, respectively. If $\text{A}$ beats $\text{B}$ by $... 761 views 1 answers 1 votes 761 views Bottle$1$contains a mixture of milk and water in$7: 2$ratio and Bottle$2$contains a mixture of milk and water in$9:4\$ ratio. In what ratio of volumes should the li...
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}