Question

The number of integer pairs (x,y) satisfy the equation `x(x+1)=2^y` is?

Answer 1

If `x` is an intger then `x(x+1)` is a product of two consecutive integers . One of which is odd and other is even. Again the `2^y`,where `y` is an integer has the values `2,4,8,16...`.

So `2^y` has no multiple of odd intger other than 1.

Hence the given relation `x(x+1)=2^y` is satisfied only when `x=1 and y=1`.

So the number of required intger pairs is ONE.