Hello all:
I ran into this problem on the GMATPrep test.
What is the greatest prime factor of 4^17 - 2^28?
A. 2
B. 3
C. 5
D. 7
E. 11
This was a tough one because had never dealt with adding and subtracting numbers w/exponents before. I converted 4^17 to 2^34 and I was stumped from there. My natural instinct had me choose A. However, the OA is D. I tried to work through the problem to see exactly why it is D.
By quickly analyizing the patterns of the powers of 2, I could see that every 4th power of 2 ends in the units digit of 6 (ie, 2^4 =16, 2^8 = 256), and every 2nd power of ends in a units digit of 4 (ie, 2^2 = 4, 2^6 = 64). Therefore I assumed that 2^34 (4^17) would end in the units digit of 4, and 2^28 would end in the units digit of 6, the difference of which would have a units digit of 8.
That is as far as I got and I could only eliminate 5 from the answers.
Could anyone help me out with this?
GMATPrep Exponent Problem
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4^17 - 2^28 =
2^34 - 2^28 =
//factor out 2^28
2^28(2^6 - 1)=
2^28(63)
the greatest prime factor of 2^28 is 2, but the greatest prime factor of 63 is 7
D is the answer.
2^34 - 2^28 =
//factor out 2^28
2^28(2^6 - 1)=
2^28(63)
the greatest prime factor of 2^28 is 2, but the greatest prime factor of 63 is 7
D is the answer.
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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.mvshah0101 wrote:Hello all:
I ran into this problem on the GMATPrep test.
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
Note that the common factor in each term is 2^28; thus, the expression can be simplified as follows:
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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4^17 - 2^28 = (2²)^17 - 2^28mvshah0101 wrote: What is the greatest prime factor of 4^17 - 2^28?
A. 2
B. 3
C. 5
D. 7
E. 11
= 2^34 - 2^28
= 2^28(2^6 - 1)
= 2^26(64- 1)
= (2^26)(63)
= (2^26)(3)(3)(7)
So, the PRIME factors are 2, 3, and 7
Answer: D
Cheers,
Brent