MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Menu
pankaj gupta Grade: 12
        

THE NUMBER OF ORDERED PAIRS OF INTEGERS (x,y) SATISFYING THE EQUATION x^2+6x+y^2

6 years ago

Answers : (1)

SAGAR SINGH - IIT DELHI
879 Points
										

Dear student,


Rather than solving this question directly, i will let u know how these questions are solved taking some different example..


The pairs (x,y) of positive integers such that x^2-y^2 = 275 correspond 1-to-1 to pairs of positive factors (a,b) of 275 with a > b. Indeed, given (x,y) positive with x^2-y^2 = 275, we factor:

(x+y)(x-y) = 275

Clearly x > y, as x^2 = y^2+275 > y^2. So x+y > x-y > 0, and letting a = x+y, b = x-y, we have the desired pair of factors.

Conversely, factor 275 = ab, with a, b positive and a > b. Let x = (a+b)/2, y = (a-b)/2. These are positive integers since a and b must both be odd and hence their sum and difference must be even. Check that, indeed, x^2-y^2 = 275. Furthermore, check that these two maps are inverse to each other.

So now we just need to find the number of pairs of factors of 275. Well, let's just factor 275.

275 = 5^2 * 11

So the number of factors of 275 is the number of factors of 5^2 times the number of factors of 11, which is 3*2 = 6. We're looking for certain ordered pairs, so divide by 2, and we get 3 pairs. We can list these out: (275,1), (55,5), (25,11). Use the correspondence above to find the values for x and y, e.g. x = (275+1)/2, y = (275-1)/2.


 
















Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.


All the best.


Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.


Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar  respectively : Click here to download the toolbar..


 


Askiitians Expert


Sagar Singh


B.Tech, IIT Delhi


sagarsingh24.iitd@gmail.com















6 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details