SOLUTION: A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495? (Order does not matter. So, the pair "16 and 9" is the same

Algebra ->  Square-cubic-other-roots -> SOLUTION: A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495? (Order does not matter. So, the pair "16 and 9" is the same       Log On


   



Question 1084247: A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495? (Order does not matter. So, the pair "16 and 9" is the same as "9 and 16".)
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A number is called a perfect square if it is the square of an integer. How many pairs of perfect squares differ by 495?
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495 = 247 + 248
248^2 - 247^2 = 495

Answer by ikleyn(52926) About Me  (Show Source):
You can put this solution on YOUR website!
.
1.  x%5E2+-+y%5E2 = (x+y)*(x-y).


2.  495 = 5*99 = 3%5E2%2A5%2A11.


Therefore, each presentation x%5E2+-+y%5E2 = 495 corresponds to one of the system


a) x + y = 495.
   x - y =   1;

b) x + y = 165;    (= 495/3)
   x - y     3;

c) x + y = 55,     ( = 495/9)
   x - y =  9;

d) x + y = 99,     (= 495/5)
    x - y = 5;

e) x + y = 45,     (=495/11)
   x - y = 11;     

and so on . . . 

Did you get the idea ?

     The number of different presentations x%5E2-y%5E2 = 495 is equal to the number of natural divisors of the number 495.

The number of different divisors of the number 495 we can determine based on its prime decomposition

        495 = 3%5E2%2A5%2A11.     (1)

It is equal to (2+1)*(1+1)*(1+1) = 3*2*2 = 12.


The numbers in parentheses is 1 (one) PLUS the index of the prime divisors in decomposition (1).

      This problem is of the level "Math+", which means the level slightly above the standard school math.

      So I assume that your level corresponds to the level of the problem and do not go in further explanations.