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

Question

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

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

chetan2u · Accepted Answer

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

Tmoni26 · Answer

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

Kingsman · Answer

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.

davedekoos · Answer

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)

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)^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.

Answer: C

Does that help?

Cheers,

sq01 · Answer

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 )

SeregaP · Answer

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

Push2018 · Answer

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.

Abhishek009 · Answer

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

ScottTargetTestPrep · Answer

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

KanishkM · Answer

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

Basshead · Answer

Given:  27^3 - 9^3 - 3^6

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

3^9 - 3^6 - 3^6

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

3^6 * 5^2

Largest prime factor is 5. Answer is C.

Vaishnavi12345 · Answer

Hey, why can't we subtract 3^6 from 3^6 directly, which will leave us with 27^3 only? then the answer will be 3.

Abhishek009 · Answer

27^3−9^3−3^6

Or,  3^9−3^6−3^6

Or,  3^6(3^3−1−1)

Or,  3^6(3^3−2)

Or,  3^6(27−2)

Or,  3^6*25

Or,  3^6*5*5 , Thus largest prime factor is 5, Answer must be (C)

ScottTargetTestPrep · Answer

Response:

I think you interpreted 3^9 - 3^6 - 3^6 as 3^9 - (3^6 - 3^6), in which case we would indeed subtract 3^6 from 3^6. However, the expression 3^9 - 3^6 - 3^6 is actually equivalent to 3^9 + (-3^6) + (-3^6). It may be helpful to rewrite the expression as - 3^6 + 3^9 - 3^6 to see why it would be wrong to subtract 3^6 from 3^6 and reduce the expression to 3^9.

lmcclellan18 · Answer

Would someone be willing to help me here? Trying to sharpen my understanding of what when wrong;

1. Translated bases to 3

27^3 = (3^3)^3 = 3^9
9^3 = (3^2)^3 = 3^6
3^6 = Leave as is.

2. Lined up Equation and Solved

3^9 - 3^6 - 3^6
3^3 - 3^6
3^-3

3. Concluded still only 3 factors of 3, thus 3 remains largest prime factor.

chetan2u · Answer

Portion 2 is wrong. You cannot add powers this way. Please go through exponents. 
 https://gmatclub.com/forum/math-number-theory-88376.html#p666609 
 3^9-3^6-3^6=3^9-2*3^6=3^6(3^3-2)=3^6*(27-2)=3^65^2

So 5 is the largest prime factor.

mariaraashidkoul · Answer

The factorization you've done is wrong. It leads to 3^6 - 3^5 - 3^5 instead.
The correct factorization would be 3^6(3^3-1-1) which leads for the highest prime factor to be 5.

swapnils10 · Answer

Simplify the given equation.

(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

So the answer is C

Hope it helps !

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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?

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