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

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.

Answer: C

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 6x� + 2x³ + 8x² to get 2x²(3x³ + x + 4)
- Factoring x² + 5x + 6 to get (x + 2)(x + 3)
- Factoring x² - y² 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 54² - 53²
If we recognize that this is a difference of squares in the form x² - y², we can factor it to get:
54² - 53² = (54 + 53)(54 - 53) = (107)(1) = 107

Likewise, the expression 2¹�� - 2�� is no different from x¹�� - x��
x¹�� - x�� = x��(x� - 1) in the exact same way that 2¹�� - 2�� = 2��(2� - 1)

Cheers,
Brent

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.

Final Answer: C

GMAT Assassins aren’t born, they’re made,
Rich