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Math Expert V
Joined: 02 Sep 2009
Posts: 64174
What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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4
17 00:00

Difficulty:   45% (medium)

Question Stats: 58% (01:13) correct 42% (01:15) wrong based on 500 sessions

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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.

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Math Expert V
Joined: 02 Aug 2009
Posts: 8626
Re: What is the largest prime factor of 27^3−9^3−3^6?  [#permalink]

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2
4
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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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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1
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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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 $$(3^3)^2[3^3-1-1] = (3^3)^2* = 3^6*5^2$$

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

Does that help?

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

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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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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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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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1
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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1
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

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

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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.

$$27^3−9^3−3^6$$
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?  