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# Which number is a factor of 1001^32-1

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Which number is a factor of 1001^32-1  [#permalink]

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Updated on: 07 May 2013, 22:17
3
10
00:00

Difficulty:

95% (hard)

Question Stats:

44% (01:55) correct 56% (01:52) wrong based on 133 sessions

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Which of the following is a factor of $$1001^{32}-1$$

A. 768
B. 819
C. 826
D. 858
E. 924

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Originally posted by mau5 on 07 May 2013, 11:57.
Last edited by mau5 on 07 May 2013, 22:17, edited 3 times in total.
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Re: Which number is a factor of 1001^19-1  [#permalink]

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07 May 2013, 15:05
5
2
using prime factoring, we can break down 1001

1001 is a factor of 7 (700+280+21), and we're left with 143
143 is a factor of 11, and we're left with 13
13 is a factor of, well, 13

thus 1001^19 contains NINETEEN 7's, 11's and 13's, and is thus perfectly divisible by all of these
.... which means that 1001^19 -1 is NOT divisible by any of 7, 11, or 13

by process of elimination, B, C, and E are out, since they're all multiples of 7 (and therefore not valid divisors of our original number)
similarly, D is out since it's a multiple 11

--> A is the winner by POE, since it's the only surviving answer
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Re: Which number is a factor of 1001^19-1  [#permalink]

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07 May 2013, 15:35
1
This is not a good question. What DDB1981 says is correct, the numbers must be co-prime => no common prime factor. So if one of the options has 7,11 or 13 as factor then the option is wrong.

However this is not enough to say that A is a factor.

Infact 768 is NOT a factor of $$1001^{19}-1$$. I calculated it using this. Press mod to see the remainder.
This question has no correct answer, what is the source?
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Re: Which number is a factor of 1001^19-1  [#permalink]

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07 May 2013, 20:04
Hi,

I tried calculating it from the link provided by you and the no gets perfectly divided by 768.

The explanation provided is perfect. That is how we attack such question. Two consecutive numbers do not have any common factor. I am also for A.

Excellent explanation. I heard that 1001 is a number to be remembered as it is a product of 3 consecutive prime numbers 7, 11, 13 and is sometimes used in exams.

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Re: Which number is a factor of 1001^32-1  [#permalink]

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08 May 2013, 01:42
1
Given $$1001^{32}-1$$--> The next consecutive integer is $$1001^{32}$$ which can be factorized as $$(7*11*13)^{32}$$. Note that any 2 consecutive integers are co-prime, i.e. they don't have any common factor other than 1. As 7,11, and 13 are all factors of $$1001^{32}$$, thus $$1001^{32}-1$$ will have NO common factors. From the given options, we have to look for a number, which is not a factor of any of the three primes. Only A satisfies the criteria.
A.

+1 to DDB1981
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Re: Which number is a factor of 1001^19-1  [#permalink]

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23 Feb 2014, 08:27
Zarrolou wrote:

Infact 768 is NOT a factor of $$1001^{19}-1$$. I calculated it using this. Press mod to see the remainder.
This question has no correct answer, what is the source?

Zarrolou,

Shouldnt we actually be concerned about $$1001^{32}-1$$. Why have you calculated for $$1001^{19}-1$$ ? Thank you.
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Re: Which number is a factor of 1001^19-1  [#permalink]

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23 Feb 2014, 08:35
HKHR wrote:
Zarrolou wrote:

Infact 768 is NOT a factor of $$1001^{19}-1$$. I calculated it using this. Press mod to see the remainder.
This question has no correct answer, what is the source?

Zarrolou,

Shouldnt we actually be concerned about $$1001^{32}-1$$. Why have you calculated for $$1001^{19}-1$$ ? Thank you.

_________
It's a typo.
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Re: Which number is a factor of 1001^32-1  [#permalink]

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10 Jun 2014, 05:13
1
Hi Bunnel,

Could you please comment if my understanding of the problem is correct.

a^n+b^n is divisible by a+b when n is even,and a^n-b^n is always divisible by a-b

〖1001〗^32-1=〖1001〗^32-1^32 is divisible by 1001+1=1002=2.3.167 and 1001-1=1000=2^3.5^3

From this limited information we know that 〖1001〗^32-1 is definitely a factor of 〖2.3.167.2〗^3.5^3 or any combinations of these integers

By implementing this information we can see that option B,C,D,E doesnot contain 167 as prime factor,there fore A must be an option as 768=(2^4.3).2^4

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Re: Which number is a factor of 1001^32-1  [#permalink]

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10 Oct 2014, 04:07
1
using a^2 - b^2 = (a+b) (a-b)
we get 1000* 1002^5 = 2^3 * 5^3 * 2^5 * 3^5 * 167^5
now check which options are divisible by 2,3,5 or 167

A. 768
B. 819
C. 826
D. 858
E. 924

A has 3 * 2^8 clear winner
for B-E it will take 20 seconds more...hope this approach is clear
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Which number is a factor of 1001^32-1  [#permalink]

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10 Oct 2014, 07:27
A id the correct option as per above explation
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Re: Which number is a factor of 1001^32-1  [#permalink]

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09 Mar 2018, 13:00
Top Contributor
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 squares
And 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))

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.

Cheers,
Brent
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Re: Which number is a factor of 1001^32-1 &nbs [#permalink] 09 Mar 2018, 13:00
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