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# How many positive integers less than 28 are prime numbers

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Intern
Joined: 06 Mar 2012
Posts: 32
How many positive integers less than 28 are prime numbers [#permalink]

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17 Mar 2012, 16:57
2
3
00:00

Difficulty:

85% (hard)

Question Stats:

56% (02:21) correct 44% (02:10) wrong based on 210 sessions

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How many positive integers less than 28 are prime numbers, odd multiples of 5, or the sum of a positive multiple of 2 and a positive multiple of 4?

A. 27
B. 25
C. 24
D. 22
E. 20

My thought process and correct solution:

List the prime numbers from 0-28 (3,5,7,11,13,17,19,23), odd multiples of 5 (5,15,25), and sum of a positive multiple of 2 and a positive multiple of 4 (since it has to be a sum the first integer starts at 6 onwards (6,8,10,12,14, 16,18, 20, 22, 24 and 26). The total integers add up to 8 + 3 + 11= 22
This seems rather time consuming. Is this really the quickest way to answer the question? Thanks in advance.
Intern
Joined: 16 Feb 2012
Posts: 34
Location: United States
Concentration: Entrepreneurship, Technology
GMAT 1: 690 Q47 V38
GPA: 3.7
Re: How many positive integers... [#permalink]

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17 Mar 2012, 19:29
Did not understand the third part of the question correctly. Anyways, baseline is that I followed the same approach as you mentioned above.

Created 4 Columns (actually there should be only 3), First had all the prime numbers, Second had 5's multiple, and third for the third set (which I messed up, but ) approach was similar to yours.

Will wait for some to post smarter way of tackling this.
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Intern
Joined: 03 Feb 2012
Posts: 8
Re: How many positive integers less than 28 are prime numbers [#permalink]

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18 Mar 2012, 11:05
damham17 wrote:
How many positive integers less than 28 are prime numbers, odd multiples of 5, or the sum of a positive multiple of 2 and a positive multiple of 4?

A. 27
B. 25
C. 24
D. 22
E. 20

My thought process and correct solution:

List the prime numbers from 0-28 (3,5,7,11,13,17,19,23), odd multiples of 5 (5,15,25), and sum of a positive multiple of 2 and a positive multiple of 4 (since it has to be a sum the first integer starts at 6 onwards (6,8,10,12,14, 16,18, 20, 22, 24 and 26). The total integers add up to 8 + 3 + 11= 22
This seems rather time consuming. Is this really the quickest way to answer the question? Thanks in advance.

Regarding your thought process, you missed the number "2" for your prime numbers, and you should eliminate one of the 5's (since 5 is in both the "prime number" and "odd multiples of 5" category). But other than that, I solved the problem the same way you did.
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Joined: 11 Jul 2012
Posts: 41
Re: How many positive integers less than 28 are prime numbers [#permalink]

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22 Oct 2012, 22:46
First there are only 9 integers thess than 28 whore are prime numbers (2 3 5 7 11 13 17 19 23) how come you your answers choices are all greater than 9?
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Joined: 02 Sep 2009
Posts: 46264
Re: How many positive integers less than 28 are prime numbers [#permalink]

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23 Oct 2012, 05:02
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1
Ousmane wrote:
First there are only 9 integers thess than 28 whore are prime numbers (2 3 5 7 11 13 17 19 23) how come you your answers choices are all greater than 9?

We wan to determine how many numbers less than 28 are primes OR odd multiples of 5 OR the sum of a positive multiple of 2 and a positive multiple of 4.

How many positive integers less than 28 are prime numbers, odd multiples of 5, or the sum of a positive multiple of 2 and a positive multiple of 4?

A. 27
B. 25
C. 24
D. 22
E. 20

9 prime numbers less than 28: {2, 3, 5, 7, 11, 13, 17, 19, 23}

3 odd multiples of 5: {5, 15, 25}

11 numbers which are the sum of a positive multiple of 2 and a positive multiple of 4: {6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26}

Notice, that 5 is in two sets, thus total # of integers satisfying the given conditions is 9+3+11-1=22.

Hope it's clear.
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Re: How many positive integers less than 28 are prime numbers [#permalink]

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12 Feb 2014, 21:04
What is meant by the sum of a positive multiple of 2 and a positive multiple of 4
First of all does it actually means --- the sum of a positive multiple of 2 and 4?

Multiple of 2 - 2,4,6,8,10,12,14,16,18,20,22,24,26
Multiple of 4 - 4,8,12,16,20,24

What is sum of these Didn't understand
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Re: How many positive integers less than 28 are prime numbers [#permalink]

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12 Feb 2014, 22:09
4
b2bt wrote:
What is meant by the sum of a positive multiple of 2 and a positive multiple of 4
First of all does it actually means --- the sum of a positive multiple of 2 and 4?

Multiple of 2 - 2,4,6,8,10,12,14,16,18,20,22,24,26
Multiple of 4 - 4,8,12,16,20,24

What is sum of these Didn't understand

It means you need to find those numbers which can be written as 2a + 4b where a and b are positive integers.

n = 2(a + 2b)
Note that a + 2b can take all values starting from 3
a = 1, b = 1, a+2b = 3
a = 2, b = 1, a+2b = 4
a = 1, b = 2, a+2b = 5
a= 2, b = 2, a+2b = 6
etc...
Hence the numbers which are "sum of a positive multiple of 2 and a positive multiple of 4" all are even numbers starting from 6 onwards.
n = 6, 8, 10, 12 etc

You can also do this question by figuring out the number of integers which are none of these three: not prime, not odd multiples of 5, not even (6 and above). The reason you will do that is there will be few such numbers since most numbers less than 28 are either even or prime. Also the options are close to 27 so they tell you that there are few such numbers

Start from 1.
1 - not prime, not multiple of 5, not even (6 and above)
4 - not prime, not multiple of 5, not even (6 and above)
Just focus on the odd numbers now and keep ignoring primes: 5/7 - ignore
9 - not prime, not multiple of 5, not even (6 and above)
11/13/15/17/19 - Ignore
21 - not prime, not multiple of 5, not even (6 and above)
23/25 - Ignore
27 - not prime, not multiple of 5, not even (6 and above)

So there are 5 numbers that do not fall in any of these categories out of a total of 27 numbers which are less than 28.

This gives us 27 - 5 = 22 relevant numbers

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Re: How many positive integers less than 28 are prime numbers [#permalink]

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13 Dec 2017, 07:51
damham17 wrote:
How many positive integers less than 28 are prime numbers, odd multiples of 5, or the sum of a positive multiple of 2 and a positive multiple of 4?

A. 27
B. 25
C. 24
D. 22
E. 20

We can start by listing the number of prime numbers less than 28:

2, 3, 5, 7, 11, 13, 17, 19, 23

There are 9 prime numbers less than 28.

Next we list the odd multiples of 5 less than 28:

5, 15, 25

There are 3 odd multiples of 5. However, since 5 is also a prime number and we don't want to count it twice, we can remove that number from this list and thus there are only 2 odd multiples of 5 left.

Finally, we need to determine the number of sums from adding positive multiples of 2 and positive multiples of 4 that are less than 28:

Since the smallest positive multiple of 2 is 2 and the smallest multiple of 4 is 4, we see that all even numbers from 2 + 4 = 6 to 26, inclusive, will be a sum of a multiple of 2 and (a multiple of) 4. This is because any even number greater than or equal to 6 can be expressed as 4 + some even number. There are (26 - 6)/2 + 1 = 11 even numbers from 6 to 26, inclusive.

Thus, we have a total of 9 + 2 + 11 = 22 numbers that fit the given criteria.

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How many positive integers less than 28 are prime numbers [#permalink]

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28 Dec 2017, 06:00
Prime numbers < 28 = {2,3,5,7,11,13,17,19,23} = total 9

odd multiples of 5 < 28 = {5,10,15} => excluding 5 as prime number counted already = total 2

integer = sum of multiple of 2 and sum of multiple of 4 < 28 => basically all positive even numbers(except 2,4) < 28 (as 2 and 4 cannot be expressed as sum of multiple of 2 and multiple of 4) = total = (13 - 2) = 11

total = 9 + 2 + 11 = 22 => (D)
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Joined: 02 Aug 2009
Posts: 5911
Re: How many positive integers less than 28 are prime numbers [#permalink]

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28 Dec 2017, 06:27
damham17 wrote:
How many positive integers less than 28 are prime numbers, odd multiples of 5, or the sum of a positive multiple of 2 and a positive multiple of 4?

A. 27
B. 25
C. 24
D. 22
E. 20

My thought process and correct solution:

List the prime numbers from 0-28 (3,5,7,11,13,17,19,23), odd multiples of 5 (5,15,25), and sum of a positive multiple of 2 and a positive multiple of 4 (since it has to be a sum the first integer starts at 6 onwards (6,8,10,12,14, 16,18, 20, 22, 24 and 26). The total integers add up to 8 + 3 + 11= 22
This seems rather time consuming. Is this really the quickest way to answer the question? Thanks in advance.

Another way is to find THOSE that don't fit the bill..

1) the sum of a positive multiple of 2 and a positive multiple of 4 MEANS 2x+4y but x and y cannot be 0
so all even numbers other than 2 and 4 fall into this category..
2) Prime numbers - the above 2 also gone and other prime number
3) All multiples of 5 gone

so we are looking at 1,4 and possible multiples of ODD numbers..
a) if 3 - $$3^2, 3^3$$ and 3*7
b) if 5 - None
c) if 7 - cannot be more than 7*3 as 28 is 7*4

so final list 1,4,9,27,21 - 5 numbers

ans 27-5=22
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Re: How many positive integers less than 28 are prime numbers   [#permalink] 28 Dec 2017, 06:27
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