## Arithmetic - Properties of Numbers: If n = ...

##### This topic has expert replies
Junior | Next Rank: 30 Posts
Posts: 10
Joined: 18 Apr 2012
Thanked: 1 times

### Arithmetic - Properties of Numbers: If n = ...

by wied81 » Sat Apr 28, 2012 5:37 pm

00:00

A

B

C

D

E

## Global Stats

This is from OG 13 #117 in Problem Solving:

If N = 3^8 - 2^8, which of the following is NOT a factor of n?

A) 97

B) 65

C) 35

D) 13

E) 5

OA: C

The book posts a pretty obscure way to solve, would like to hear others opinions on this question.

Community Manager
Posts: 1060
Joined: 13 May 2011
Location: Utrecht, The Netherlands
Thanked: 318 times
Followed by:52 members
by neelgandham » Sat Apr 28, 2012 6:18 pm
If N = 3^8 - 2^8, which of the following is NOT a factor of n?

3^8 - 2^8 = (3^4-2^4)(3^4+2^4) - a^2 -b^2 = (a+b)*(a-b)
3^8 - 2^8 = (3^2-2^2)(3^2+2^2)(3^4+2^4) - a^2 -b^2 = (a+b)*(a-b)
3^8 - 2^8 = (3-2)(3+2)(3^2+2^2)(3^4+2^4)
3^8 - 2^8 = (1)(5)(13)(97), so 97,65(13*5),13,5 are factors of 3^8 - 2^8. The only number left is 35, which isn't a factor of 3^8 - 2^8
Anil Gandham
Welcome to BEATtheGMAT | Photography | Getting Started | BTG Community rules | MBA Watch
Check out GMAT Prep Now's online course at https://www.gmatprepnow.com/

Master | Next Rank: 500 Posts
Posts: 425
Joined: 08 Dec 2010
Thanked: 56 times
Followed by:7 members
GMAT Score:690
by LalaB » Sun Apr 29, 2012 9:27 am
3^8 - 2^8=(3^4-2^4)(3^4+2^4)=
=(81-16)(81+16)=65*97 (a and b are out)

65*97=5*13*97 (d and E are out)
answ is C
Happy are those who dream dreams and are ready to pay the price to make them come true.(c)

In order to succeed, your desire for success should be greater than your fear of failure.(c)

### GMAT/MBA Expert

GMAT Instructor
Posts: 6365
Joined: 25 Apr 2015
Location: Los Angeles, CA
Thanked: 43 times
Followed by:26 members
by [email protected] » Sat Jun 27, 2015 5:21 am
wied81 wrote:This is from OG 13 #117 in Problem Solving:

If N = 3^8 - 2^8, which of the following is NOT a factor of n?

A) 97

B) 65

C) 35

D) 13

E) 5

OA: C

The book posts a pretty obscure way to solve, would like to hear others opinions on this question.
Solution:

It is very unlikely that a problem would require us to calculate 3^8 or 2^8, so we should approach this problem not as an arithmetic question but as an algebraic one.

The first thing we should recognize is that we are being tested on the algebraic factoring technique called the "difference of squares." Recall that the general form of the difference of squares is:

x^2 - y^2 = (x + y)(x - y)

Similarly, we can treat 3^8 - 2^8 as a difference of squares, which can be expressed as:

n = (3^4 + 2^4)(3^4 - 2^4)

We can further factor 3^4 - 2^4 as an additional difference of squares, which can be expressed as:

(3^2 + 2^2)(3^2 - 2^2)

This finally gives us:

n = 3^8 - 2^8 = (3^4 + 2^4)(3^2 + 2^2)(3^2 - 2^2)

The numbers are now easy to calculate:

n = (81 + 16)(9 + 4)(9 - 4)

n = (97)(13)(5)

We are being asked which of the answer choices is NOT a factor of n, which we have determined to be equal to the product (97)(13)(5). So we must find the answer choice that does not evenly divide into (97)(13)(5).

We immediately see that 97, 13 and 5 are all factors of (97)(13)(5).

This leaves us with 65 and 35. Notice that (97)(13)(5) = (97)(65). Thus, 65 also is a factor of n. Only 35 is not.

### GMAT/MBA Expert

GMAT Instructor
Posts: 15867
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1267 members
GMAT Score:770
by [email protected] » Sat Jun 27, 2015 8:42 am
I want to add something about the concept of algebraic manipulations without variables:

The GMAT loves to test our algebra skills (like factoring), HOWEVER many students associate algebra with variables only, so they don't see that we can apply algebraic principles to numbers as well (which should make sense, since those variables are, indeed, representing numbers).

So, for example, many students are fine with the following:
- Factoring 6x + 3 to get 3(2x + 1)
- Factoring 6xâ�µ + 2xÂ³ + 8xÂ² to get 2xÂ²(3xÂ³ + x + 4)
- Factoring xÂ² + 5x + 6 to get (x + 2)(x + 3)
- Factoring xÂ² - yÂ² to get (x + y)(x - y)

On the GMAT, we need to recognize that we can also factor expressions that have no variables.
So, for example, a GMAT question might ask us to evaluate 54Â² - 53Â²
If we recognize that this is a difference of squares in the form xÂ² - yÂ², we can factor it to get:
54Â² - 53Â² = (54 + 53)(54 - 53) = (107)(1) = 107

Likewise, the expression 2Â¹â�°â�° - 2â�¹â�¶ is no different from xÂ¹â�°â�° - xâ�¹â�¶
xÂ¹â�°â�° - xâ�¹â�¶ = xâ�¹â�¶(xâ�´ - 1) in the exact same way that 2Â¹â�°â�° - 2â�¹â�¶ = 2â�¹â�¶(2â�´ - 1)

Cheers,
Brent

GMAT Instructor
Posts: 2630
Joined: 12 Sep 2012
Location: East Bay all the way
Thanked: 625 times
Followed by:118 members
GMAT Score:780
by [email protected] » Mon Jun 29, 2015 3:36 pm
Here's an approach that doesn't involve difference of squares.

If a number divides by 65, it must divide by 13 and by 5. So D and E *CANNOT* be the answers: if the number is not divisible by 5, it is also not divisible by 65. (Same logic for 13 and 65.) Now that we know the answer divides by 5 and 13, it must also divide by 65, so B CANNOT be the answer.

Now we'll consider A and C. Since we know our number divides by 5, it will divide by 35 if and only if our number also divides by 7. (This is because 35 = 5 * 7).

At this point, we recall that 3â�¸ - 2â�¸ will divide by 7 if 3â�¸ and 2â�¸ have the same remainder when divided by 7. 2â�¸ = 256, which has a remainder of 4 when divided by 7. 3â�¸ = 6561, which has a remainder of 2.

So our number doesn't divide by 7, and hence can't divide by 35. We're done, and we don't need to bother testing A.

Legendary Member
Posts: 518
Joined: 12 May 2015
Thanked: 10 times
by nikhilgmat31 » Thu Jul 02, 2015 4:24 am
Factoring Formula - xÂ² - yÂ² to get (x + y)(x - y) is the key to solve this question.

### GMAT/MBA Expert

Elite Legendary Member
Posts: 10392
Joined: 23 Jun 2013
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:508 members
GMAT Score:800

### Re: Arithmetic - Properties of Numbers: If n = ...

by [email protected] » Fri Apr 23, 2021 2:56 pm
Hi All,

We’re told that N = 3^8 – 2^8. We’re asked which of the follow 5 numbers is NOT a factor of N. The GMAT would NEVER require that you calculate that overall value, so there must be a way to ‘simplify’ that equation. Since it includes Exponents and SUBTRACTION of two values raised to the SAME EVEN power, we should be on the lookout for Classic Quadratics…

You’re probably familiar with X^2 – Y^2 (since that is one of the common “Classic” Quadratics… X^2 – Y^2 = (X + Y)(X – Y)). Similar Quadratics exist for X^4 – Y^4, X^6 – Y^6 and X^8 – Y^8. Quadratic rules apply whether there are variables or numbers involved, so we can replace the X and Y with the “3” and “2”, respectively in the given calculation…

3^8 – 2^8 =
(3^4 + 2^4)(3^4 – 2^4)

We can then ‘factor down’ the 2nd part of that step…

(3^4 + 2^4)(3^4 – 2^4) =
(3^4 + 2^4)(3^2 + 2^2)(3^2 – 2^2)

And then ‘factor down’ the 3rd part of that step…

(3^4 + 2^4)(3^2 + 2^2)(3 + 2)(3 - 2) =

This gives us…
(81+16)(9+4)(5)(1)
(97)(13)(5)(1)

With these results, we can clearly eliminate Answers A, D and E. By multiplying the 13 and 5, we get (13)(5) = 65, so that is ALSO a factor – and we can eliminate Answer B.