For how many positive integer values of n is 2^8 + 2^{11} + 2^n equal to the square of an … — GMAT

For how many positive integer values of $n$ is $2^8 + 2^{11} + 2^n$ equal to the square of an integer?

A
Zero
OneCevap
C
Two
D
Three
E
Four

Cevap

There is exactly 1 positive integer value of

n

(specifically,

n = 12

The option stating 'One' is correct because factoring out

2^8

yields

2^8(9 + 2^{n-8})

. For this product to be a perfect square,

9 + 2^{n-8}

must equal

k^2

for some integer

k

. Rewriting this as

(k-3)(k+3) = 2^{n-8}

forces both factors to be powers of 2 whose difference is 6. The unique solution to

2^b - 2^a = 6

a = 1

and

b = 3

, which gives

n - 8 = 4

, so

n = 12

. No other positive integers

n \le 8

produce a square.

Adım Adım Çözüm

Analyze the expression for

n > 8

by factoring out

2^8

2^8 + 2^{11} + 2^n = 2^8 (1 + 2^3 + 2^{n-8}) = 2^8 (9 + 2^{n-8})

Since

2^8 = (2^4)^2

is already a perfect square, the entire expression is a square if and only if

9 + 2^{n-8}

is a perfect square.

Set

9 + 2^{n-8} = k^2

for some integer

k > 3

and factor using prime power properties.

k^2 - 9 = 2^{n-8} \implies (k - 3)(k + 3) = 2^{n-8}

The difference of squares allows us to express the product of two integers as a power of 2.

Solve for the prime factors of the terms

(k-3)

and

(k+3)

Let

k - 3 = 2^a

and

k + 3 = 2^b

where

a + b = n - 8

and

b > a

. Subtracting the two equations gives

(k + 3) - (k - 3) = 2^b - 2^a = 6 \implies 2^a(2^{b-a} - 1) = 6 = 2^1 \cdot 3

The prime factorization of 6 uniquely dictates that

2^a = 2^1 \implies a = 1

, and

2^{b-a} - 1 = 3 \implies 2^{b-1} = 4 \implies b = 3

Determine

n

and test values of

n \le 8

Since

a = 1

and

b = 3

n - 8 = 1 + 3 = 4 \implies n = 12

. Testing

n \le 8

reveals no other squares (e.g., for

n=8

2^8(1+8+1)=10 \cdot 2^8

, not a square; for

n=3

2312 = 2^3 \cdot 17^2

, not a square).

This confirms that

n = 12

is the unique positive integer solution.

Anahtar Kavram

Prime factorization of differences of squares and prime power analysis

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