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

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Intern
Joined: 22 Jun 2010
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How many integers less than 1000 have no factors (other than [#permalink]

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23 Aug 2010, 13:57
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How many integers less than 1000 have no factors (other than 1) in common with 1000 ?

a. 400
b. 399
c. 410
d. 420
[Reveal] Spoiler: OA
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Joined: 02 Sep 2009
Posts: 44400

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23 Aug 2010, 14:17
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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 $$999-(499+199-99)=400$$.

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03 Sep 2010, 05:24
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
Joined: 02 Sep 2009
Posts: 44400

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03 Sep 2010, 05:40
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.
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03 Sep 2010, 09:01
I agree the answers are basically the same
Director
Joined: 23 Apr 2010
Posts: 559

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11 Jan 2011, 02:55
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
Joined: 02 Sep 2009
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11 Jan 2011, 03:03
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.
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23 Jul 2011, 06:27
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?
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Re: How many integers less than 1000 have no factors (other than [#permalink]

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07 Jan 2013, 12:18
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.
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Re: How many integers less than 1000 have no factors (other than [#permalink]

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07 Jan 2013, 20:54
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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.
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Re: How many integers less than 1000 have no factors (other than [#permalink]

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01 Apr 2013, 03:25
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
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Re: How many integers less than 1000 have no factors (other than [#permalink]

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02 Apr 2013, 12:30
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.
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03 Oct 2013, 14:09
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 $$999-(499+199-99)=400$$.

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
Joined: 02 Sep 2009
Posts: 44400

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04 Oct 2013, 01:01
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 $$999-(499+199-99)=400$$.

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.
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Re: How many integers less than 1000 have no factors (other than [#permalink]

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26 Jul 2017, 06:14
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

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Re: How many integers less than 1000 have no factors (other than [#permalink]

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09 Aug 2017, 13:30
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

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Re: How many integers less than 1000 have no factors (other than [#permalink]

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15 Aug 2017, 10:22
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

Re: How many integers less than 1000 have no factors (other than   [#permalink] 15 Aug 2017, 10:22
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