GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

It is currently 23 Feb 2020, 19:29

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

How many integers less than 1000 have no factors (other than

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Intern
Intern
avatar
Joined: 22 Jun 2010
Posts: 27
How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 23 Aug 2010, 12:57
3
37
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

50% (02:21) correct 50% (02:19) wrong based on 382 sessions

HideShow timer Statistics

How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 61403
Re: good one  [#permalink]

Show Tags

New post 23 Aug 2010, 13:17
8
13
mehdiov wrote:
How many integers less than 1000 have no factors(other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420


First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

\(1000=2^3*5 ^3\) so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - \(\frac{998-2}{2}+1=499\);
Multiples of 5 in the range 0-1000, not inclusive - \(\frac{995-5}{5}+1=199\);
Multiples of both 2 and 5, so multiples of 10 - \(\frac{990-10}{10}+1=99\).

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to \(999-(499+199-99)=400\).

Answer: A.
_________________
General Discussion
Manager
Manager
avatar
Joined: 30 Aug 2010
Posts: 80
Location: Bangalore, India
Re: good one  [#permalink]

Show Tags

New post 03 Sep 2010, 04:24
Bunuel,

Yes -- we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5 = which done not have 2/5 as a factor.

For this we can USE the VENN diagram technique as shown below

The integers <= 1000 divigible by 2 = 1000/2 = 500, but = 499 if 1000 is excluded
The integers <= 1000 divigible by 5 = 1000/5 = 200, but = 199 if 1000 is excluded
The integers <= 1000 divigible by 10(2*5) = 1000/10 = 100, but = 99 if 1000 is excluded


hence, integers that r divisible by 2only and 5only = 500+200-100 (or 499+199-99 if 1000 excluded)= 600 (599 if 1000 is excluded)

so the answer is 1000-600 (or 999 - 599) = 400.

am i correct.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 61403
Re: good one  [#permalink]

Show Tags

New post 03 Sep 2010, 04:40
muralimba wrote:
Bunuel,

Yes -- we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5 = which done not have 2/5 as a factor.

For this we can USE the VENN diagram technique as shown below

The integers <= 1000 divigible by 2 = 1000/2 = 500, but = 499 if 1000 is excluded
The integers <= 1000 divigible by 5 = 1000/5 = 200, but = 199 if 1000 is excluded
The integers <= 1000 divigible by 10(2*5) = 1000/10 = 100, but = 99 if 1000 is excluded


hence, integers that r divisible by 2only and 5only = 500+200-100 (or 499+199-99 if 1000 excluded)= 600 (599 if 1000 is excluded)

so the answer is 1000-600 (or 999 - 599) = 400.

am i correct.


Yes, it's correct. Basically the same way as used in my post.
_________________
Intern
Intern
avatar
Joined: 22 Jun 2010
Posts: 27
Re: good one  [#permalink]

Show Tags

New post 03 Sep 2010, 08:01
I agree the answers are basically the same
Director
Director
avatar
Joined: 23 Apr 2010
Posts: 502
Re: good one  [#permalink]

Show Tags

New post 11 Jan 2011, 01:55
Bunuel, why can't we simply divide 1000 by 2 to find the number of multiples of 2? My reasoning is that every second number is a multiple of 2 so there must be exactly 500 numbers.

Thanks.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 61403
Re: good one  [#permalink]

Show Tags

New post 11 Jan 2011, 02:03
nonameee wrote:
Bunuel, why can't we simply divide 1000 by 2 to find the number of multiples of 2? My reasoning is that every second number is a multiple of 2 so there must be exactly 500 numbers.

Thanks.


There are 100/2=500 multiple of 2 in the range 1-1000 INCLUSIVE. As we need numbers LESS than 1000 which are also multiples of 2 then we should subtract 1 from that number. So there are total of 500-1=499 multiples of 2 in the range 0-1000, not inclusive.
_________________
Intern
Intern
avatar
Joined: 23 Jul 2011
Posts: 1
Re: good one  [#permalink]

Show Tags

New post 23 Jul 2011, 05:27
Hi!
I have a book with this question and it says, that the correct answer 401...i see that there is no such answers in your questions...so i really confused..can somebody explain why it can be 401? or it is a 100% mistake?
Intern
Intern
avatar
Joined: 13 Oct 2012
Posts: 37
Concentration: General Management, Leadership
Schools: IE '15 (A)
GMAT 1: 760 Q49 V46
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 07 Jan 2013, 11:18
The question asks for the number of integers less than 1000 and other than 1.
Isnt one included in the 400 integers that you are claimimg to be the answer?
Answer should be 399 if we exclude 1.
Please correct me in case i missed something.
Veritas Prep GMAT Instructor
User avatar
V
Joined: 16 Oct 2010
Posts: 10113
Location: Pune, India
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 07 Jan 2013, 19:54
1
2
rohantiwari wrote:
The question asks for the number of integers less than 1000 and other than 1.
Isnt one included in the 400 integers that you are claimimg to be the answer?
Answer should be 399 if we exclude 1.
Please correct me in case i missed something.


The question does not ask you to exclude 1.

Every positive integer less than 1000 has one common factor with 1000. What is it? It is 1.
1 is a common factor between any two positive integers.

If the question were: How many positive integers less than 1000 have no factors in common with 1000 ?
Then the answer would be 0. There are no positive integers which have no common factors with 1000. All the positive integers have a common factor and that is 1. But the question wants to know the number of positive integers which have no common factor other than 1 (1 will always be a common factor). Basically, it is looking for positive integers which are co-prime with 1000.
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Manager
Manager
avatar
Joined: 26 Feb 2013
Posts: 51
Concentration: Strategy, General Management
GMAT 1: 660 Q50 V30
WE: Consulting (Telecommunications)
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 01 Apr 2013, 02:25
all odd numbers excluding odd multiples of 5 have only 1 as common factor with 1000.
hence 500 odd numbers-((995-5)/10)+1)= 400
Manager
Manager
User avatar
Joined: 07 May 2012
Posts: 52
Location: United States
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 02 Apr 2013, 11:30
1
consider integers between 1 and 100 - half of them are even - hence 50 integers are multiples of 2 ( which also includes even multiples of 5) + 10 odd multiples of 5 = 60
Hence 40 integers that are not multiples of 2 and/or 5 -
hence considering integers between 1 and 1000 - there are 40*10 = 400 integers which do not have common multiple with 1000 other than 1.
_________________
Jyothi hosamani
Intern
Intern
avatar
B
Joined: 08 Jul 2012
Posts: 9
Re: good one  [#permalink]

Show Tags

New post 03 Oct 2013, 13:09
Bunuel wrote:
mehdiov wrote:
How many integers less than 1000 have no factors(other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420


First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

\(1000=2^3*5 ^3\) so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - \(\frac{998-2}{2}+1=499\);
Multiples of 5 in the range 0-1000, not inclusive - \(\frac{995-5}{5}+1=199\);
Multiples of both 2 and 5, so multiples of 10 - \(\frac{990-10}{10}+1=99\).

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to \(999-(499+199-99)=400\).

Answer: A.




What about the prime numbers Bunuel ?? For ex : 7. Neither its a multiple of 2, nor 5 and it does not has any common factors with 1000 (except 1)
So, shouldn't the answer include prime numbers between 1-999 as well. And if YES, how do we calculate the number of primer numbers from 1-999 ???
Plz clarfily.

Thanks.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 61403
Re: good one  [#permalink]

Show Tags

New post 04 Oct 2013, 00:01
sumitchawla wrote:
Bunuel wrote:
mehdiov wrote:
How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420


First of all it should be "how many positive integers less than 1000 have no factors (other than 1) in common with 1000", as if we consider negative integers answers will be: infinitely many.

\(1000=2^3*5 ^3\) so basically we are asked to calculate the # of positive integrs less than 1000, which are not multiples of 2 or/and 5.

Multiples of 2 in the range 0-1000, not inclusive - \(\frac{998-2}{2}+1=499\);
Multiples of 5 in the range 0-1000, not inclusive - \(\frac{995-5}{5}+1=199\);
Multiples of both 2 and 5, so multiples of 10 - \(\frac{990-10}{10}+1=99\).

Total # of positive integers less than 1000 is 999, so # integers which are not factors of 2 or 5 equals to \(999-(499+199-99)=400\).

Answer: A.




What about the prime numbers Bunuel ?? For ex : 7. Neither its a multiple of 2, nor 5 and it does not has any common factors with 1000 (except 1)
So, shouldn't the answer include prime numbers between 1-999 as well. And if YES, how do we calculate the number of primer numbers from 1-999 ???
Plz clarfily.

Thanks.


We counted multiples of 2 or 5 in the range 0-1000, not inclusive and then subtracted that from total number of integers in the range 0-1000. The number we get contains all numbers which are not multiples of 2 or 5, thus all primes (apart from 2 and 5) in that range too.

Hope it's clear.
_________________
Director
Director
User avatar
D
Affiliations: IIT Dhanbad
Joined: 13 Mar 2017
Posts: 722
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 26 Jul 2017, 05:14
mehdiov wrote:
How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420


Firstly question should mention +ve integers, which is not mentioned in this case.

1000 = 2^3 * 5^3

Total +ve integers (less than 1000) divisible by 2 = 1000/2 - 1 = 499
Total +ve integers(less than 1000) divisble by 5 = 1000/5 -1 = 199
Total +ve integers(less than 1000) divisble by both 2 and 5 (chk divisibility by 10) = 1000/10 -1 = 99

So total +ve integers (less than 1000) divisible by either 2 or 5 or both = 499 + 199 - 99 = 599

Integers less than 1000 have no factors (other than 1) in common with 1000 = 999 - 599 = 400

Answer A
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2801
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 09 Aug 2017, 12:30
mehdiov wrote:
How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420


Since 1,000 breaks down to prime factors of twos and fives, we need to find all the numbers less than 1,000 that do not contain those factors. To do so, let’s find all the numbers less than 1000 that contain factors of two’s and five’s. Note that all even numbers (multiples of 2) and all multiples of 5 must be accounted for.

Number of even numbers less than 1000:

(998 - 2)/2 + 1 = 499

Number of multiples of five less than 1000:

(995 - 5)/5 + 1 = 199

We must find the double-counted numbers, also called overlap numbers, which are numbers that are multiples of both 2 and 5. To find the overlap, we need to determine the number of multiples of 5 and 2 (or of 10) less than 1000:

(990 - 10)/10 + 1 = 99

Thus, the number of multiples of 2 or multiples of 5 less than 1000 is:

499 + 199 - 99 = 599

Finally, the number of numbers less than 1000 that ARE NOT multiples of 2 or 5 is:

999- 599 = 400

Answer: A
_________________

Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com
TTP - Target Test Prep Logo
181 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager
Manager
User avatar
B
Joined: 31 Oct 2016
Posts: 101
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 15 Aug 2017, 09:22
The answer is pretty simple, in my opinion. You don't need almost any calculations. 1000 has only 2 distinct factors - 2 and 5. It is obvious that there are 500 odd and even numbers from 1 to 1000. Exclude 1000 (even number) you will have 500 odd numbers and 499 even numbers. And there are 200 multiples of 5 between 1 and 1000 (1000/5). 100 of them are odd numbers (5, 15, 25 etc.) and 100 of them are even numbers (10, 20, 30 etc.). Exclude 1000 - you will have 100 and 99 respectively. (p.s. you don't even need to exclude 1000 since we will not count 1000 anyway).

Hence, we will just need to exclude odd multiples of 5 from a list of odd numbers from 1 to 999. We need only yellow space :)

Image
Manager
Manager
avatar
B
Joined: 29 May 2017
Posts: 164
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: How many integers less than 1000 have no factors (other than  [#permalink]

Show Tags

New post 02 Jan 2020, 00:22
VeritasKarishma wrote:
rohantiwari wrote:
The question asks for the number of integers less than 1000 and other than 1.
Isnt one included in the 400 integers that you are claimimg to be the answer?
Answer should be 399 if we exclude 1.
Please correct me in case i missed something.


The question does not ask you to exclude 1.

Every positive integer less than 1000 has one common factor with 1000. What is it? It is 1.
1 is a common factor between any two positive integers.

If the question were: How many positive integers less than 1000 have no factors in common with 1000 ?
Then the answer would be 0. There are no positive integers which have no common factors with 1000. All the positive integers have a common factor and that is 1. But the question wants to know the number of positive integers which have no common factor other than 1 (1 will always be a common factor). Basically, it is looking for positive integers which are co-prime with 1000.


Hi......can you explain why the multiples of 10 are being subtracted?

Thanks!
GMAT Club Bot
Re: How many integers less than 1000 have no factors (other than   [#permalink] 02 Jan 2020, 00:22
Display posts from previous: Sort by

How many integers less than 1000 have no factors (other than

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne