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

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