Question

The number of pairs (x, y) of positive integers such that
$\sqrt{x +} + \sqrt{y} = \sqrt{1980}$

B) 4
C) 5
D) 6
A) 3
2​

Answer 1

Answer:

Step-by-step explanation:

Correct option is

C

7

xy

x+y

​

=  

2007

1

​

 

⇒xy−2007(x+y)=0

Adding 2007  

2

 to both sides, we get

xy−2007(x+y)+2007  

2

=2007  

2

 

⇒(x−2007)(y−2007)=2007  

2

 

Let x−2007=A and y−2007=B

The equation becomes AB=2007  

2

 

Number of solutions of above equation is equal to number of factors of 2007  

2

 

2007  

2

=3  

4

×223  

2

 

Hence, number of factors of 2007  

2

 is (4+1)(2+1)=15

In one case A=B=2007

Of the remaining 14 cases, half of the case A>B and remaining half A<B

Accordingly we get 7 cases, where x<y.

Video Explanation

Permutations and Combinations- Problem 33 and its Solution