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C
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Difficulty:
55%
(hard)
Question Stats:
58% (01:55) correct
42%
(01:57)
wrong
based on 77
sessions
History
Date
Time
Result
Not Attempted Yet
What is the greatest prime factor of \(2^{10}*5^4-2^{13}*5^2+2^{14}\)?
- A. 2
- B. 3
- C. 7
- D. 11
- E. 13
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22 Apr 2020, 03:38
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Rdbasak
The greats prime factor of \(2^{10}*5^{14}-2^{13}*5^{2}+2^{14}\) is 699,383. There must be a typo in the question.
22 Apr 2020, 04:07
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Bunuel
So, the question should have been What is the greatest prime factor of \(2^{10}*5^4-2^{13}*5^2+2^{14}\)?
\(2^{10}*5^4-2^{13}*5^2+2^{14}=(2^5)^2*(5^2)^2-2*(2^{5}*5^2)*2^7+(2^7)^2=(2^5*5^2-2^7)^2=2^{10}(5^2-2^2)^2=2^{10}*(21)^2=2^{10}*(3*7)^2\).
Answer: C.
Or:
\(2^{10}*5^4-2^{13}*5^2+2^{14}=2^{10}5^2(5^2-2^{3})^2+2^{14}=2^{10}5^2*17+2^{14}=2^{10}(5^2*17+2^{4})=2^{10}*441=2^{10}*3^2*7^2\).
Answer: C.
Or:
\(2^{10}*5^4-2^{13}*5^2+2^{14}=2^{10}*5^4-2^{13}(5^2-2)=2^{10}*5^4-2^{13}*23=2^{10}(5^4-2^{3}*23)=2^{10}*3^2*7^2\).
Answer: C.
