What is the greatest prime factor of 4^17-2^28?
a)2
b)3
c)5
d)7
e)11
Exponentials and prime factors -OG
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To answer this we must find the prime factorization of 4^17 - 2^28Kuros wrote:What is the greatest prime factor of 4^17 - 2^28?
a)2
b)3
c)5
d)7
e)11
To do this, we'll apply to algebraic factoring techniques (and some exponent rules).
Since 4 is not prime, let's first take 4^17 and replace 4 with 2^2
When we do this, we get (2^2)^17 - 2^28
We can now apply the Power of a Power Rule to rewrite this as 2^34 - 2^28
From here, let's factor out 2^28 to get 2^28(2^6 - 1)
2^6 evaluates to be 64, so we get: 2^28(64 - 1)
This equals 2^28(63)
We can find the prime factorization of 63 to write this as (2^28)(3)(3)(7)
So, 4^17 - 2^28 = (2^28)(3)(3)(7), which means the greatest prime factor is 7
Answer: D
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Hi Kuros,
When dealing with 'exponent rule' questions, you often have to make sure that you're comparing "like" base values.
Here, we have 4^17 and 2^28.... so we're starting with DIFFERENT bases. We can 'rewrite' 4 as (2^2) though, which gives us....
(2^2)^17 - 2^28
Now that we're taking "a power to a power", we multiply the 2 and the 17....
2^34 - 2^28
From here, we can factor out a common element: 2^28
(2^28)(2^6 - 1)
2^6 is relatively easy to calculate.... 64
(2^28)(64 - 1)
(2^28)(63)
(2^28)(7)(3)(3)
Thus, the greatest prime factor here is 7
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
When dealing with 'exponent rule' questions, you often have to make sure that you're comparing "like" base values.
Here, we have 4^17 and 2^28.... so we're starting with DIFFERENT bases. We can 'rewrite' 4 as (2^2) though, which gives us....
(2^2)^17 - 2^28
Now that we're taking "a power to a power", we multiply the 2 and the 17....
2^34 - 2^28
From here, we can factor out a common element: 2^28
(2^28)(2^6 - 1)
2^6 is relatively easy to calculate.... 64
(2^28)(64 - 1)
(2^28)(63)
(2^28)(7)(3)(3)
Thus, the greatest prime factor here is 7
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
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- Jay@ManhattanReview
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Hi Kuros,Kuros wrote:What is the greatest prime factor of 4^17-2^28?
a)2
b)3
c)5
d)7
e)11
4^17 - 2^28 = (2^2)^17 - 2^28
=> 2^34 - 2^28
=> 2^28(2^6 - 1)
=> 2^28(64 -1 )
=> 2^28(63)
=> (2^28)*(3^2)*(7)
We see that there are three prime factors in (2^28)*(3^2)*(7); these are 2, 3, and 7. The greatest among them is 7.
The correct answer: D
Hope this helps!
Relevant book: Manhattan Review GMAT Number Properties Guide
-Jay
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We need to determine the greatest prime factor of 4^17 - 2^28. We can start by breaking 4^17 into prime factors.Kuros wrote:What is the greatest prime factor of 4^17-2^28?
a)2
b)3
c)5
d)7
e)11
4^17 = (2^2)^17 = 2^34
Now our equation is as follows:
2^34 - 2^28
We see that 2^28 is a common factor in both terms, so we factor it out:
2^28(2^6 - 1)
2^28(64 - 1)
2^28(63)
2^28 x 9 x 7
2^28 x 3^2 x 7
We see that the greatest prime factor must be 7.
Answer: D
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