Problem Solving

If x and y are integers and

15^x + 15^x+1 \4^y = 15^y,

what is the value of x?

2
3
4
5
Cannot be determined

This is the way it is solved, but I don't understand all of it, especially how do you get "15^x(1+15)???

15^x + 15^x+1\4y = 15^y

(15^x + 15^x+1) = 15^y(4^y)
[15^x + 15^x(15^1)] = 15^y(4^y)
(15^x )(1 + 15) = 15^y(4^y)
(15^x)(16) = 15^y(4^y)
(3^x)(5^x)(2^4) = (3^y)(5^y)(22^y)

Since both sides of the equation are broken down to the product of prime bases, the respective exponents of like bases must be equal.


2y = 4 so y = 2.
x = y so x = 2.

The correct answer is A.

15^x + 15^x+1\4y = 15^y

(15^x + 15^x+1) = 15^y(4^y)
[15^x + 15^x(15^1)] = 15^y(4^y)

taking 15^x as common on LHS we get
(15^x )(1 + 15) = 15^y(4^y)
(15^x)(16) = 15^y(4^y)
(3^x)(5^x)(2^4) = (3^y)(5^y)(22^y)