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21 Feb 2020, 05:37
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For which integer n is \(2^8 + 2^{11} + 2^n\) is a perfect square?
A. 4
B. 6
C. 8
D. 11
E. 12
Are You Up For the Challenge: 700 Level Questions
A. 4
B. 6
C. 8
D. 11
E. 12
Are You Up For the Challenge: 700 Level Questions
21 Feb 2020, 05:54
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Bunuel
For which integer n is \(2^8 + 2^{11} + 2^n\) is a perfect square?
A. 4
B. 6
C. 8
D. 11
E. 12
A. 4
B. 6
C. 8
D. 11
E. 12
\(2^8 + 2^{11} + 2^n\) must be aperfect square for some value given among options
Let's check options
A) 4, i.e. Expression becomes \(2^8 + 2^{11} + 2^4 = 2^4(2^4+2^7+1) = 16*(16+128+1) = 16*145\) NOT a perfect square
B) 6, i.e. Expression becomes \(2^8 + 2^{11} + 2^6 = 2^6(2^2+2^5+1) = 2^6*(4+32+1) = 2^6*37\) NOT a perfect square
C) 8, i.e. Expression becomes \(2^8 + 2^{11} + 2^8 = 2^8(1+2^3+1) = 2^8*(10) \) NOT a perfect square
D) 11, i.e. Expression becomes \(2^8 + 2^{11} + 2^{11} = 2^8(1+2^3+2^3) = 2^8*(17) \) NOT a perfect square
E) 12, i.e. Expression becomes \(2^8 + 2^{11} + 2^{12} = 2^8(1+2^3+2^4) = 2^8*(1+8+16) = 2^8*25\) a perfect square
General Discussion
21 Feb 2020, 05:55
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\(2^{8} ( 1+8+ 2^{n—8} ) = 16^{2} ( 9+2^{n—8} ) \)
—> Eliminate A,B and C.
D) 11 --> \(16^{2} ( 9+2^{11–8})= 16^{2} *17 \)
(Not perfect square)
E) 12 --> \(16^{2} (9+2^{12–8}) = 16^{2}*5^{2} \)
The answer choice E is correct.
Posted from my mobile device
—> Eliminate A,B and C.
D) 11 --> \(16^{2} ( 9+2^{11–8})= 16^{2} *17 \)
(Not perfect square)
E) 12 --> \(16^{2} (9+2^{12–8}) = 16^{2}*5^{2} \)
The answer choice E is correct.
Posted from my mobile device
