mau5 wrote:
Which of the following is a factor of \(1001^{32}-1\)
A. 768
B. 819
C. 826
D. 858
E. 924
The key to answering this question is to recognize that 1001^32 − 1 is a
difference of squaresAnd so it 1001^16 - 1
And 1001^18 - 1
etc
1001^32 − 1 = (1001^16 + 1)(1001^16 - 1)
= (1001^16 + 1)(1001^8 + 1)(1001^8 - 1)
= (1001^16 + 1)(1001^8 + 1)(1001^4 + 1)(1001^4 - 1)
= (1001^16 + 1)(1001^8 + 1)(1001^4 + 1)(1001^2 + 1)(1001^2 - 1)
= (1001^16 + 1)(1001^8 + 1)(1001^4 + 1)(1001^2 + 1)(1001 + 1)(1001 - 1)
Now let's evaluate some of the NICE parts.
= (1001^16 + 1)(1001^8 + 1)(1001^4 + 1)(1001^2 + 1)(
1002)(
1000= (1001^16 + 1)(1001^8 + 1)(1001^4 + 1)(1001^2 + 1)(
(2)(3)(167))(
(2)(2)(2)(3)(3)(3))
Now check the answer choices....
A. 768 = (2)(2)(2)(2)(2)(2)(2)(2)(3) = (2^8)(3)
Hmmm, it
looks like we might not have enough 2's in the factorization of 1001^16 - 1 in order for 768 to be a factor.
However, if we recognize that (1001^16 + 1), (1001^8 + 1), (1001^4 + 1), and (1001of ^2 + 1) are all EVEN numbers, we can see that we have enough two's in the factorization of 1001^16 - 1 for 768 to be a factor.
Answer: A
Cheers,
Brent
_________________
Test confidently with gmatprepnow.com