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The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is

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New post 21 Nov 2014, 01:42
5
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A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

58% (02:33) correct 42% (02:30) wrong based on 168 sessions

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The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)

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New post Updated on: 24 Nov 2014, 21:17
[quote="PareshGmat"]The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)[/quo

Originally posted by Ashishmathew01081987 on 21 Nov 2014, 07:52.
Last edited by Ashishmathew01081987 on 24 Nov 2014, 21:17, edited 1 time in total.
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New post 21 Nov 2014, 09:22
Let's simplify both expressions first:

\((3^13+3^10)=3^13(3^3+1)=3^10*28=3^10*2*2*7\)
\((3^13-3^10)=3^13(3^3-1)=3^10*26=3^10*2*13\)

Clearly, the common factor is \(3^10*2\), but none of the answer choices matches it. So, we need to try to simplify the answer choices. Once we get to D, we get \(3^11-9^5=3^11-3^10=3^10(3-1)=3^10*2\)

Sorry for my formulas being off - new to this formatting thing.
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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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New post 21 Nov 2014, 12:56
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Ashishmathew01081987 wrote:

\(3^{13} + 9^5 = 3^{13} + 3^{10} = 3^{23}\)
\(3^{13} - 9^5 = 3^{13} - 3^{10} = 3^3\)


hi this is incorrect

\(a^{b} . a^{c} = a^{b+c}\)

\(also,\) \(a^{b}. a^{-c} = a^{b-c}\)

PareshGmat wrote:
The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)


\((3^{13} + 9^5)\) = \(3^{13} + 3^{10}\)
\(=3^{10} (3^{3} +1 )
=3^{10}28
= 2^{2}. 7 . 3^{10}\) -------------------------1)
\((3^{13} - 9^5)\) = \(3^{13} - 3^{10}\)
\(= 3^{10} (3^{3} - 1 )
=3^{10}26
=2 . 13 . 3^{10}\) -----------------------------2)

as can be seen the highest common factor of 1 and 2 is \(2 . 3^{10}\)

option D = \(3^{11} - 9^5\)
\(= 3^{11} - 3^{10}

=3^{10} (3-1)

= 2 . 3^{10}\)

hence answer is D
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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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New post 23 Nov 2014, 21:53
manpreetsingh86 wrote:
Ashishmathew01081987 wrote:

\(3^{13} + 9^5 = 3^{13} + 3^{10} = 3^{23}\)
\(3^{13} - 9^5 = 3^{13} - 3^{10} = 3^3\)


hi this is incorrect

\(a^{b} . a^{c} = a^{b+c}\)

\(also,\) \(a^{b}. a^{-c} = a^{b-c}\)

PareshGmat wrote:
The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)


\((3^{13} + 9^5)\) = \(3^{13} + 3^{10}\)
\(=3^{10} (3^{3} +1 )
=3^{10}28
= 2^{2}. 7 . 3^{10}\) -------------------------1)
\((3^{13} - 9^5)\) = \(3^{13} - 3^{10}\)
\(= 3^{10} (3^{3} - 1 )
=3^{10}26
=2 . 13 . 3^{10}\) -----------------------------2)

as can be seen the highest common factor of 1 and 2 is \(2 . 3^{10}\)

option D = \(3^{11} - 9^5\)
\(= 3^{11} - 3^{10}

=3^{10} (3-1)

= 2 . 3^{10}\)

hence answer is D



Thanks for pointing out my mistake. That was a careless error :oops: +1 Kudos for identifying it :lol:
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New post 25 Nov 2014, 11:51
PareshGmat wrote:
The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)



It is D. Factor out 3^10 from both expressions and solve. one expression gives you 28, and the other gives you 26. GCD of these two is 2.
Therefore 3^10 *2.
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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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New post 02 Apr 2016, 00:44
PareshGmat wrote:
The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)


Hi,
Since we are to find HCF, we will have to simplify the terms..
\((3^{13} + 9^5)\) = \((3^{13} + 3^{10})\)
=>\(3^{10}*(3^3 + 1)\) =\(3^{10}*28\)


\((3^{13} - 9^5)\) = \((3^{13} - 3^{10})\)
=>\(3^{10}*(3^3 - 1)\) =\(3^{10}*26\)

so HCF = 3^{10}*2

Let see the choices..


A: \(3^{13}\)... Eliminate

B: \(3^{12} - 9^5\)..simplify

C: \(2*3^{12}\).. . Eliminate

D: \(3^{11} - 9^5\)..simplify

E: \(3^{11}\)... Eliminate

lets see B and D


B: \(3^{12} - 9^5\)

\((3^{13} - 9^5)\) = \((3^{12} - 3^{10})\)
=>\(3^{10}*(3^2 - 1)\) =\(3^{10}*8\).. eliminate

D.
\((3^{11} - 9^5)\) = \((3^{11} - 3^{10})\)
=>\(3^{10}*(3 - 1)\) =\(3^{10}*2\)

ans D
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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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New post 19 Jul 2017, 00:54
PareshGmat wrote:
The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)


\((3^{13} + 9^5)\)
= \((3^{13} + 3^{10})\)
= \(3^{10}(3^{3} + 1)\)
= \(3^{10}(3^{3} + 1)\)
= \(3^{10}(7*4)\)

\((3^{13} - 9^5)\)
= \((3^{13} - 3^{10})\)
= \(3^{10}(3^{3} - 1)\)
= \(3^{10}(3^{3} - 1)\)
= \(3^{10}(13*2)\)

Highest common factor = \(2*3^{10}\)


Here most difficult part s that the options are not clear and the highest common factor has been hidden in the complex form.
Lets now start analyzing the options.

OPTION A : \(3^{13}\)
OPTION B : \(3^{12} - 9^5\) = \(3^{10}(3^{2} - 1\) = \(8*3^{10}\)
OPTION C :\(2*3^{12}\)
OPTION D : \(3^{11} - 9^5\) = \(3^{10}(3 - 1)\) = \(2*3^{10}\)
OPTION E : \(3^{11}\)


Answer D

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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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New post 31 Oct 2018, 18:23
PareshGmat wrote:
The highest common factor of \((3^{13} + 9^5)\) and \((3^{13} - 9^5)\) is

A: \(3^{13}\)

B: \(3^{12} - 9^5\)

C: \(2*3^{12}\)

D: \(3^{11} - 9^5\)

E: \(3^{11}\)


Let’s first simplify each of the numbers whose GCF we wish to determine:

3^13 + 9^5 = 3^13 + 3^10 = 3^10(3^3 + 1) = 3^10 x 28 = 2^2 x 3^10 x 7^1

3^13 - 9^5 = 3^13 - 3^10 = 3^10(3^3 - 1) = 3^10 x 26 = 2^1 x 3^10 x 13^1

The GCF is 2^1 x 3^10. We can easily eliminate answer choices A, C, and E. Simplifying answer choice D, we have:

3^11 - 9^5 = 3^11 - 3^10 = 3^10(3 - 1) = 3^10 x 2^1

Answer: D
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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is   [#permalink] 31 Oct 2018, 18:23
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