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What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ?

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Rdbasak
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What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   Updated on: 22 Apr 2020, 04:07
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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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C

Originally posted by Rdbasak on 22 Apr 2020, 03:29.
Last edited by Bunuel on 22 Apr 2020, 04:07, edited 1 time in total.
Edited the question.
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Bunuel
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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   22 Apr 2020, 03:38
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Rdbasak wrote:
What is the greatest prime factor of \(2^{10}*5^{14}-2^{13}*5^{2}+2^{14}\)?


    A. 2
    B. 3
    C. 7
    D. 11
    E. 13


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.
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Bunuel
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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   22 Apr 2020, 04:07
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Bunuel wrote:
Rdbasak wrote:
What is the greatest prime factor of \(2^{10}*5^{14}-2^{13}*5^{2}+2^{14}\)?


    A. 2
    B. 3
    C. 7
    D. 11
    E. 13


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.


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.
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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   22 Apr 2020, 10:02
\(=2^{10}(5^4 -5^2*2^3+2*4)\\
\)
\(= 2^{10}(625-200+16)\)
\(= 2^{10}(441)\)
\(= 2^{10}* 7*7*9\)
thus greatest prime divisor is 7
C
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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   24 Apr 2020, 03:41
Rdbasak wrote:
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


Hello!
I found an alternate quick method which can be done using the units digit of the numbers.
Units digit of each number shall be:-
2^10*5^4 - 2^13*5^2 + 2^14= 4 x 5 - 2 x 5 + 4
It adds upto 14.

7 is the closest answer which i found!

OA-: Option C

Let me know if it does not suffice!

Thank you!
Cheers :thumbsup:

Regards,
Raunak Damle
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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   24 Apr 2020, 09:01
Expert Reply
Rdbasak wrote:
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


Factoring, we have:

2^10(5^4 - 2^3 * 5^2 + 2^4)

2^10(625 - 200 + 16)

2^10(441)

2^10(9 x 49)

2^10 x 3^2 x 7^2

Thus, the largest prime factor is 7.

Answer: C
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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink] New post   24 Apr 2020, 09:16
Rdbasak wrote:
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


Asked: What is the greatest prime factor of \(2^{10}*5^4-2^{13}*5^2+2^{14}\)?

\(2^{10}*5^4-2^{13}*5^2+2^{14}\)
\(= 2^{10}(5^4 - 2^3*5^2 + 2^4)\)
\(= 2^{10}(625 - 200 + 16) \)
\(= 2^{10}(441)\)
\(= 2^{10}*3^2*7^2\)

IMO C

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Re: What is the greatest prime factor of 2^10*5^4 - 2^13*5^2 + 2^14 ? [#permalink]
24 Apr 2020, 09:16
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