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What is the largest prime factor of 27^3−9^3−3^6?

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What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 19 Feb 2016, 03:06
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 19 Feb 2016, 03:55
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Bunuel wrote:
What is the largest prime factor of 27^3−9^3−3^6?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for correct solution.


Hi,
the equation tells us that all are with base 3..
We should simplify equation...
\(27^3−9^3−3^6=(3^3)^3-(3^2)^3-3^6...
=3^9-2*3^6=3^6(3^3-2)=3^6*25..\)
so 5 is the greatest prime factor

C
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 19 Feb 2016, 08:58
I am probably wrong but I will my solution out there and someone will tell me where I fell down

So 27^3 - 9^3 - 3^6, get them into bases of 3s
27^3 = (3^3)^3
9^3 = (3^2)^3 = (3^3)^2
3^6 = (3^3)^2

So we have (3^3)^3 -(3^3)^2 - (3^3)^2
Factor out 3^3, we have 3^3 (3^3 - 3^2 - 3^2) ----> 3^3 (27 -9-9) ----> (27) (9) -->
Highest prime factor is 3
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post Updated on: 19 Feb 2016, 20:39
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One vote for C. Here is my sol:
27^3−9^3−3^6 = 3^9 - 3^6 - 3^6 = 3^6( 3^3-1-1) = 3^6 (27-2) = 3^6 * 5^2.

Two prime factors are 3 and 5. Therefore, max prime factor is 5.
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 19 Feb 2016, 12:28
Tmoni26 wrote:
I am probably wrong but I will my solution out there and someone will tell me where I fell down

So 27^3 - 9^3 - 3^6, get them into bases of 3s
27^3 = (3^3)^3
9^3 = (3^2)^3 = (3^3)^2
3^6 = (3^3)^2

So we have (3^3)^3 -(3^3)^2 - (3^3)^2
Factor out 3^3, we have 3^3 (3^3 - 3^2 - 3^2) ----> 3^3 (27 -9-9) ----> (27) (9) -->
Highest prime factor is 3



Hi Tmoni,

Your approach was right, but where you "fell down" is when you factored out the \(3^3\).

Factoring out \(3^3\) from \((3^3)^3-(3^3)^2-(3^3)^2\) will give you \((3^3)[(3^3)^2-(3^3)-(3^3)]\)

Consider replacing \(3^3\) with \(x\), then the expression would look like \(x^3-x^2-x^2\)
Now if you factor out an \(x\) what happens? You get \(x(x^2-x-x)\)

Remember, \(x^2 = x*x\), so \((3^3)^2 = (3^3)*(3^3)\). When you factor out one \(3^3\), you still have one \(3^3\) left over, not \(3^2\)

So in fact you could have factored out \((3^3)^2\) and gotten \((3^3)^2[3^3-1-1] = (3^3)^2*[25] = 3^6*5^2\)

Now the expression is broken down into its prime factors and we can see that the greatest prime factor is 5.

Answer: C

Does that help?



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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 19 Feb 2016, 14:01
What is the largest prime factor of 27^3−9^3−3^6

Step #1: Convert everything to base 3--> (3^3)^3-(3^2)^3-3^6
Step #2: Reduce using rules of powers (rules: (x^a)^b = x^(a*b) and x^a+x^b=x^(a+b)) --> 3^9-3^6-3^6
Step #3: Simplify by factoring --> 3^6(3^3-1-1) =3^6(3^3-2)
Step #4: Larget prime of this form--> 3^6(3^3-2) =3^6(27-2)= 3^6(25) -->Prime Factor =3( Prime Factor 5)
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 18 Mar 2017, 13:42
let's make the equation to the base of 3: 3^9-2*3^6=3^6(3^3-2)=3^6*25
hence 5 is the highest prime
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 01 Apr 2017, 07:49
27^3-9^3-3^6
= 3^3(3) - 3^2(3) - 3^6
= 3^9 - 3^6 - 3^6
= 3^9 - 2(3^6)
=3^6 (3^3-2)
=3^6 (27-2)
=3^6 (25)
=3^6 (5^2)

The largest prime factor is 5.
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 01 Apr 2017, 23:02
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Bunuel wrote:
What is the largest prime factor of \(27^3−9^3−3^6\)?

A. 2
B. 3
C. 5
D. 7
E. 11


\(27^3−9^3−3^6\)

=>\(3^9 − 3^6 − 3^6\)

=>\(3^6 ( 3^3 − 1 − 1 )\)

=>\(3^6 ( 3^3 − 2 )\)

=>\(3^6*25\)

Thus, the largest prime factor will be 5, answer must be (C) 5
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 09 Jul 2018, 18:58
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Bunuel wrote:
What is the largest prime factor of \(27^3−9^3−3^6\)?

A. 2
B. 3
C. 5
D. 7
E. 11


The key to solving this problem is to express each of the terms with a base of 3. Doing this, we have:

(3^3)^3 - (3^2)^3 - 3^6

3^9 - 3^6 - 3^6

3^6(3^3 - 1 - 1)

3^6(25) = 3^6 x 5^2

Answer: C
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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New post 16 Feb 2019, 19:37
Bunuel wrote:
What is the largest prime factor of \(27^3−9^3−3^6\)?

A. 2
B. 3
C. 5
D. 7
E. 11

Kudos for correct solution.


\(27^3−9^3−3^6\)
Can be written as

\(3^9−3^6−3^6\)

Take \(3^6\), common to get

\(3^6\) \((3^3 -1 -1)\)

\(3^6\) * 25

5 -> largest prime factor
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Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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Re: What is the largest prime factor of 27^3−9^3−3^6?   [#permalink] 14 Mar 2020, 23:30

What is the largest prime factor of 27^3−9^3−3^6?

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