Prove that it is impossible to find integers x and y such th

Prove that it is impossible to find integers x and y such that 2^x + 6 = 8y + 5.

Solution

Prove that it is impossible to find integers x and y such that

2^x + 6 = 8y + 5.

Rearranging : 2^x = 8y -1

for x> 0 LHS is even and RHS is odd.

for x=0 2^0 = 8y -1

8y = 2----> y = 1/4 ( which is not an integer)

for x<0 2^x is a fraction and RHS 8y-1 is an integer

So, there are no integer solutions for x and y for the equation 2^x + 6 = 8y + 5.