If x and y are positive integers such that 2^x - 2^y = 1920, what is the value of x + y? — GRE General Test

If $x$ and $y$ are positive integers such that $2^x - 2^y = 1920$ , what is the value of $x + y$ ?

The correct value is 18.

Factoring

2^y

from

2^x - 2^y

yields

2^y(2^{x-y} - 1)

. Factoring 1920 as

2^7 \times 15

allows us to uniquely match the power-of-two factor

2^y = 2^7

(

y = 7

) and the odd factor

2^{x-y} - 1 = 15

(

x - y = 4 \implies x = 11

). Summing

x

and

y

gives

11 + 7 = 18

Factor out the smaller power of 2 from the expression

2^y(2^{x-y} - 1) = 1920

Since

x

and

y

are positive integers and

2^x - 2^y > 0

, we know

x > y

. Factoring

2^y

separates the even power of 2 component from an odd component.

Determine the prime factorization of 1920 into a power of 2 and an odd integer

1920 = 128 \times 15 = 2^7 \times 15

Repeatedly dividing 1920 by 2 gives

1920 = 2^7 \times 15

, where 15 is an odd integer.

Equate the even and odd components

2^y = 2^7 \implies y = 7

and

2^{x-y} - 1 = 15 \implies 2^{x-y} = 16 = 2^4 \implies x - y = 4

Because

x - y \ge 1

, the quantity

2^{x-y} - 1

must be an odd integer, forcing it to equal 15 and the power of 2 factor to equal

2^7

Solve for

x

and calculate the final sum

x + y

x = 7 + 4 = 11

, so

x + y = 11 + 7 = 18

Adding

x = 11

and

y = 7

gives the required value.

Factoring exponential expressions by pulling out the common base power and matching unique prime factorizations.

Question