Question

Find the number of ordered pairs of positive integers (x y) that satisfy x^(2)+y^(2)-xy=37​

Answer 1

Given :   x² + y²  - xy  = 37

To find : ordered pairs of positive integers (x y) that satisfy

Solution:

x² + y²  - xy  = 37

=> (x - y)²  + xy = 37

=> (x - y)² = 37 - xy  

=> xy  ≤  37  

(x - y)² = 0  => xy  = 37   ( 1, 37 )

& x = y  hence no possible Solution

(x - y)² = 1   => xy  =  36      

=> x - y = 1     &  possible x , y   ( 1 , 36) , ( 2, 18) , ( 3, 12) , ( 4, 9) , ( 6 , 6)

none satisfy  x - y = 1

(x - y)² = 4   => xy  =  33   ( 1 , 33) or ( 3 , 11)

x - y = 2   hence  no possible Solution

(x - y)² = 9   => xy  =  28   ( 1 , 28) or ( 2 , 14) , ( 4, 7)

x - y = 3    hence ( 4 , 7 )  Satisfied

(x - y)² = 25   => xy  =  12   ( 1 , 12) or ( 2 , 6) , ( 3, 4)

x - y = 5   hence  no possible Solution

(x - y)² = 36   => xy  =  1   ( 1 , 1)  

x - y = 6   hence  no possible Solution

So only possible Solution

is ( 4 , 7 )  or ( 7 , 4)

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