If n is a positive integer greater than 20 such that \sqrt{2^{28} + 2^{20} + 2^n} is an in… — GMAT

If $n$ is a positive integer greater than $20$ such that $\sqrt{2^{28} + 2^{20} + 2^n}$ is an integer, what is the least possible value of $n$ ?

A
22
B
24
25Cevap
D
26
E
34

Cevap

The least possible value of

n

is 25.

The expression inside the radical must be a perfect square. Matching

2^{28} + 2^{20} + 2^n

(2^{14} + 2^{10})^2 = 2^{28} + 2 \cdot 2^{14} \cdot 2^{10} + 2^{20} = 2^{28} + 2^{25} + 2^{20}

reveals that

2^n = 2^{25}

, so

n = 25

. Since 25 is greater than 20 and smaller than the other valid solution

n = 34

, it is the least possible value.

Adım Adım Çözüm

Set up the perfect square structure for the radical expression.

For

\sqrt{2^{28} + 2^{20} + 2^n}

to be an integer, the expression

2^{28} + 2^{20} + 2^n

must equal

(2^a + 2^b)^2 = 2^{2a} + 2^{a+b+1} + 2^{2b}

for some positive integers

a > b

Expanding a binomial power of 2 generates three power-of-2 terms matching the three terms in the expression.

Analyze Case 1 where

2^{28}

corresponds to the leading term

2^{2a}

2a = 28 \implies a = 14

. The remaining terms

2^{20}

and

2^n

must correspond to

2^{2b}

and

2^{a+b+1}

Matching highest powers establishes the value of the parameter

a

Evaluate sub-cases for matching

2^{20}

Subcase 1: If

2b = 20 \implies b = 10

, then the middle term exponent is

a + b + 1 = 14 + 10 + 1 = 25

. Thus

n = 25

.
Subcase 2: If

a + b + 1 = 20 \implies 14 + b + 1 = 20 \implies b = 5

, then

2b = 10

. Thus

n = 10

Checking both term assignments determines possible integer values for

n

Analyze Case 2 where

2^{20}

corresponds to

2^{2b}

and

2^{28}

is the middle term.

2b = 20 \implies b = 10

and

a + b + 1 = 28 \implies a + 11 = 28 \implies a = 17

, then

2a = 34

. Thus

n = 34

Checking alternative middle term assignments finds all valid values for

n

Filter by constraint

n > 20

and select the minimum.

The valid values satisfying

n > 20

are

n = 25

and

n = 34

. The least possible value is 25.

The question asks specifically for the least possible value greater than 20.

Anahtar Kavram

Perfect Square Trinomial Expansion with Exponent Rules

Tahmini Süre:2m 0s

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