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

Question

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

manpreetsingh86 · Accepted Answer

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

Bunuel · Answer

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Puneet1011 · Answer

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.

mvictor · Answer

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.

Celerma · Answer

Answer is C

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"

Mathivanan Palraj · Answer

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.

stonecold · Answer

Nice Question
Here is what i did

N=2^10 
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.

dave13 · Answer

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

chetan2u · Answer

Looking at the expression, we can say for sure that 2 is a factor and 5 is not.  
Beyond this, if we have to find other factors, we will have to simplify the expression.

2^{10}*5^4 - 2^{13}*5^2 + 2^{14} 
We have  2^{10}  common in all terms of the expression, so we can factor that out.

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

2^{10}*21^2=2^{10}*3^2*7^2

Greatest prime factor is 7.

C

Crytiocanalyst · Answer

it just requires slight simplification 
=>2^10 
=>2^10*441
=>2^10*3^2*7^2

Therefore IMO C

Fdambro294 · Answer

If you look closely this is a Square of a Difference Quadratic Template:

(X - Y)^2 = (X)^2 - (2)(X)(Y) + (Y)^2

X = (2)^5 * (5)^2

Y = (2)^7

^2 =

*   =

(2)^5 * (5)^2 * (2)^5 * (5)^2 - (2) * (2)^5 * (5)^2 * (2)^7 + (2)^7 * (2)^7 =

(2)^10 * (5)^4 - (2)^13 * (5)^2 + (2)^14

Thus, using the Square of a Difference Quadratic Template, we can focus on one of the Factors (other one will be identical)

-take common factor of (2)^5

(2)^5 *

If there is a larger prime factor it will come from the expression within the brackets

(5)^2 - (2)^2 =

25 - 4 = 21 = (7) (3)

(C) 7 is the largest prime factor

Posted from my mobile device

ablatt4 · Answer

step 1: recognize that this isn't 5 terms, it is 3 terms  - +2^14

Step 2: factor out the GCF 2^10(5^4- +2^4)

Simplify 2^10(625-200+16) = 2^10=441 
just divide 441 by answer choices and if none work then 2 is the answer because we know we have 2^10

3 and 7 are both factors, and 11 and 13 are not
C

What is the greatest prime factor of 2^105^4 – 2^135^2 + 2^14?

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