Are the positive integers x and y consecutive?
(1)x^2 - y^2 = 2y + 1
(2) x^2 - xy - x = 0
The question is basically asking whether x=y+1
Statement 1)x^2 - y^2=2y+1
can be written as (x+y)(x-y)=2y+1
If we put x=y+1
, then LHS must be equal to RHS.
Equation 1 can be written, after substituting x=y+1, as (2y+1)(1)=2y+1
. They are equal. Hence x and y are consecutive.
The above equation can be equal to 0 only when x-y=1
because x is given to be positive.
Hope I am correct.
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