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Property of Numbers

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Property of Numbers

by RobertP » Thu Feb 11, 2016 12:23 pm
Hello everyone,

Can some one please help me understand the solution of the problem bellow?

Thank you!

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)

Answer: C
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by DavidG@VeritasPrep » Thu Feb 11, 2016 12:48 pm
RobertP wrote:Hello everyone,

Can some one please help me understand the solution of the problem bellow?

Thank you!

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)

Answer: C
3^8 - 2^8 is a difference of squares: it can be rewritten as (3^4)^2 - (2^4)^2.
Because 3^4 = 81 and 2^4 = 16, (3^4)^2 - (2^4)^2 = 81^2 - 16^2.
Because x^2 - y^2 = (x+y)(x-y), 81^2 - 16^2 = (81 + 16)(81 - 16) = 97 * 65.
So clearly 97 and 65 are factors.
65 = 13*5, so 97*65 = 97*13*5; Therefore13 and 5 are factors.
That leaves us with C
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by Brent@GMATPrepNow » Thu Feb 11, 2016 12:50 pm
David's solution is perfect.
I just 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
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by ceilidh.erickson » Thu Feb 11, 2016 4:40 pm
Here's a guessing strategy that you could have used, if you didn't realize that the question was testing a difference of squares:

Since the problem asks which of the following is NOT a factor of n, we can infer that the other 4 answer choices will be factors of n.

If this is the case, the answer can't possibly be E. 5 can't be the number that's not a factor, if 35 and 65 both are factors. So, eliminate E.

We can eliminate D using the same logic - 13 can't be the one non-factor if that leaves 65 (which contains a factor of 13). So, eliminate D.

If we've logically eliminated D and E, that means that 5 and 13 must be factors of n, and thus 65 must also be a factor of n. So, eliminate B.

You now have it narrowed down to a 50-50 choice between A and C. If you have to guess, that's better than guessing blindly!
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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