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

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

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21 Nov 2014, 00:42
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13
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Difficulty:

75% (hard)

Question Stats:

61% (02:01) correct 39% (02:07) wrong based on 157 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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The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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Updated on: 24 Nov 2014, 20:17
[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, 06:52.
Last edited by Ashishmathew01081987 on 24 Nov 2014, 20:17, edited 1 time in total.
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The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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21 Nov 2014, 08:22
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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21 Nov 2014, 11:56
2
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 = $$3^{11} - 9^5$$
$$= 3^{11} - 3^{10} =3^{10} (3-1) = 2 . 3^{10}$$

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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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23 Nov 2014, 20:53
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 = $$3^{11} - 9^5$$
$$= 3^{11} - 3^{10} =3^{10} (3-1) = 2 . 3^{10}$$

Thanks for pointing out my mistake. That was a careless error +1 Kudos for identifying it
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The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is  [#permalink]

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25 Nov 2014, 10:51
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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01 Apr 2016, 23:44
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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18 Jul 2017, 23:54
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 : $$3^{11}$$

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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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31 Oct 2018, 17:23
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

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Re: The highest common factor of (3^13 + 9^5) and (3^13 - 9^5) is &nbs [#permalink] 31 Oct 2018, 17:23
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