# CAT 2017 Question Paper | Quants Slot 2

###### CAT Previous Year Paper | CAT Number Theory Questions | Question 28

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Question 28 : How many different pairs (a, b) of positive integers are there such that a ≤ b and $$frac{1}{a}$ + $\frac{1}{b}$ = $\frac{1}{9}$ ? [TITA] ## 2IIM : Best Online CAT Coaching #### 2IIM's Online CAT CoachingGet CAT Last Mile Prep Course for 799 /-CAT Online Coaching ### Video Explanation ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2020Starts Sat, November 2nd, 2019 ### Explanatory Answer ##### Method of solving this CAT Question from Number Theory Given that $\frac{1}{a}$ + $\frac{1}{b}$ = $\frac{1}{9}$ Let us assume that a ≤ b. For first case, let a = b, then $\frac{1}{18}$ + $\frac{1}{18}$ = $\frac{1}{9}$ Hence$18 , 18) is one pair of possibility.
In all the cases, both a and b should be greater than 9 to satisfy the equation.
So, we can take a = 10 , 11 , 12 , ..... , 18 and check for the different possibilities.
For example, let us take a = 15
⟹ $$frac{1}{b}$ = $\frac{1}{9}$ - $\frac{1}{15}$ ⟹ $\frac{5 - 3}{45}$ = $\frac{2}{45}$ Hence a = 15 is not possible. Similarly when a = 17, 16, 14, 13, and 11 does not work either. Now let us take a = 12, ⟹ $\frac{1}{b}$ = $\frac{1}{9}$ - $\frac{1}{12}$ ⟹ $\frac{4 - 3}{36}$ = $\frac{1}{36}$ Hence, a = 12 and b = 36 is the other pair of possibility. When a = 10, ⟹ $\frac{1}{b}$ = $\frac{1}{9}$ - $\frac{1}{10}$ ⟹ $\frac{10 - 9}{90}$ = $\frac{1}{90}$ Hence, a = 10 and b = 90 is the other pair of possibility. Therefore$18,18) , (12,36) and (9,90) are the three different pairs of positive integers which satisfy the condition

The question is "How many different pairs (a, b) of positive integers are there such that a ≤ b and $$frac{1}{a}$ + $\frac{1}{b}$ = $\frac{1}{9}$ ? [TITA]" ##### Hence, the answer is 3 ###### 2IIM's Online CAT CoachingGet CAT Last Mile Prep Course for 799 /- Signup Now! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2020Enroll at 44,000/- Next Weekend Batch Starts Sat, Nov 2nd, 2019 ###### Best CAT Coaching in ChennaiRegister Online, get Rs 4000/- off Attend a Demo Class ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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