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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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What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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20 Dec 2014, 06:36
5
10
00:00

Difficulty:

65% (hard)

Question Stats:

64% (02:08) correct 36% (02:22) wrong based on 359 sessions

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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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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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20 Dec 2014, 06:58
1
1
JusTLucK04 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

2^10.5^4 - 2^13. 5^2 + 2^14

first take 2^10 and 5^2 common from first two terms

2^10.5^2(5^2-2^3) + 2^14
2^10.5^2(25-8) + 2^14
2^10(25)(17) + 2^14
now take 2^10 from both the terms

2^10((25)(17) + 2^4)
2^10(425+16)
2^10(441)
2^10 .3^2. 7^2

hence largest prime factor is 7
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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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20 Dec 2014, 08:34
JusTLucK04 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

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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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20 Dec 2014, 08:58
1
JusTLucK04 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

Easy One..

First take 2^10 as common, as shown below -
2^10(5^4 - 2^3*5^2 + 2^4)
2^10(625 - 200 6)
2^10 * 441
Find factors of 441 ie 21 * 21
The greatest prime factor is 7.
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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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25 Oct 2015, 10:16
1
time consuming, but definitely not a 700-level question.
we can factor 2^10 first, and we remain with 5^2(5^2 - 2^3)+2^4

25-8 = 17, so we have 25*17 + 16, which is 441. I did not start doing the prime factorization, since it is more time consuming than trying to divide by each option choice.
it is definitely not divisible by 11, so D is out.
Divide by 13, well, 13*3 = 39, and we are left with 51, which is not divisible by 13, E out.
Try by 7, works, so definitely the GCF is 7.
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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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26 Oct 2015, 07:00

On solving further , we arrive at 2^10 * 441 and that follows 2^10 * 7^2*3^2 so the greatest prime factor is "7"
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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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16 Mar 2016, 05:24
JusTLucK04 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

Basically you have to simplify the expression.
Take 2^10 out of the expression; you get 2^10*441=2^10*21^2
That gives 7 as the greatest prime factor.
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Re: What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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22 Nov 2016, 02:17
1
Nice Question
Here is what i did

N=2^10[625-200+16]
N=2^10*441
N=2^10*3^2*7^2
Hence 7 is the highest prime factor.
Hence C
Its just a time consuming Question.
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What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?  [#permalink]

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10 Oct 2018, 04:29
JusTLucK04 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

Bunuel shouldnt initiall question look like this $$(2^{10}*5^4) - (2^{13}*5^2)+2^{14}$$

$$2^{10}*5^2(5^2-2^3)+2^{14}$$

$$2^{10}*25(25-8)+2^{14}$$

$$2^{10}*25(17)+2^{14}$$

$$2^{10}(425)+2^{14}$$

$$2^{10}(425+16)$$

$$2^{10} (441)$$

Prime factors of 441 are 3 and 7

So, the greatest prime factor is 7
What is the greatest prime factor of 2^10*5^4 – 2^13*5^2 + 2^14?   [#permalink] 10 Oct 2018, 04:29
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