retagged by
578 views

1 Answer

1 votes
1 votes
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 $:\text{A}$
edited by
Answer:

Related questions

780
views
2 answers
2 votes
go_editor asked Mar 16, 2020
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$$4/9$
501
views
1 answers
1 votes
go_editor asked Mar 13, 2020
501 views
If $a, b, c$ and $d$ are integers such that $a+b+c+d=30$ , then the minimum possible value of $( a-b )^{2}+( a-c )^{2}+( a-d)^{2}$ is $1$$2$$5$$6$
486
views
0 answers
0 votes
go_editor asked Mar 16, 2020
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...
745
views
1 answers
1 votes
go_editor asked Mar 16, 2020
745 views
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
go_editor asked Mar 16, 2020
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...