2¹�� - 2�� = 2��(2� - 1) = 2��(15) = (2��)(3)(5).Newaz111 wrote:What is the greatest prime factor of 2^100 - 2^96?
A. 2
B. 3
C. 5
D. 7
E. 11
The greatest prime factor is 5.
The correct answer is C.
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The greatest prime factor is 5.
The correct answer is C.
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2¹�� - 2�� = ?Newaz111 wrote:Can you please explain how 2^96(2^4-1) from 2^100 - 2^96. It's not clear to me.
Thank you.
Identify the GREATEST COMMON FACTOR:
(2��)(2�) - (2��)(1)
Factor out the greatest common factor:
(2��)(2� - 1)
Simplify:
(2��)(16 - 1)
(2��)(15)
(2��)(3)(5).
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Mitch's solution is perfect (as usual). I just wanted 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
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
Brent Hanneson - Creator of GMATPrepNow.com
