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# How many positive integers less than 30 are either a multiple of 2, an

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Director
Joined: 02 Sep 2016
Posts: 630
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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19 Aug 2017, 01:06
Bunuel wrote:
enigma123 wrote:
How many positive integers less than 30 are either a multiple of 2, an odd prime number, or the sum of a positive multiple of 2 and an odd prime?
(A) 29
(B) 28
(C) 27
(D) 25
(E) 23

Any idea how to solve this guys?

30 sec approach:
Any odd non-prime, greater than 1, can be obtained by the sum of an odd prime and a positive even number. So this set plus the set of odd primes basically makes the set of all odd numbers greater than 1 in the range. Now, the set of all odd numbers greater than 1 together with the set of all even numbers makes the set of all numbers from 1 to 30, not inclusive, so total of 28 numbers.

To illustrate:
# of even numbers in the range is (28-2)/2+1=14: 2, 4, 6, ..., 28;
# of odd primes in the range is 9: 3, 5, 7, 11, 13, 17, 19, 23, and 29;
# of integers which are the sum of a positive multiple of 2 and an odd prime is 5: 9=7+2, 15=13+2, 21=19+2, 25=23+2 and 27=23+4;

Total: 14+9+5=28. You can see that we have all numbers from 1 to 30, not inclusive: 2, 3, 4, 5, 6, ...., 29.

Hope it's clear.

Hello Bunuel

If in such series, we get a number that repeats in both sets. Then do we have to count it once or twice?

For example:
How many positive integers less than 20 are multiple of 2 or a multiple of 3?

Multiple of 2: 2,4,6,8,10,12,14,16,18

Multiple of 3: 3,6,9,12,15,18

So do we have to count 6, 12, and 18 once or twice?

Total would be 15 or 12 ?

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 62291
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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19 Aug 2017, 02:31
Shiv2016 wrote:
Bunuel wrote:
enigma123 wrote:
How many positive integers less than 30 are either a multiple of 2, an odd prime number, or the sum of a positive multiple of 2 and an odd prime?
(A) 29
(B) 28
(C) 27
(D) 25
(E) 23

Any idea how to solve this guys?

30 sec approach:
Any odd non-prime, greater than 1, can be obtained by the sum of an odd prime and a positive even number. So this set plus the set of odd primes basically makes the set of all odd numbers greater than 1 in the range. Now, the set of all odd numbers greater than 1 together with the set of all even numbers makes the set of all numbers from 1 to 30, not inclusive, so total of 28 numbers.

To illustrate:
# of even numbers in the range is (28-2)/2+1=14: 2, 4, 6, ..., 28;
# of odd primes in the range is 9: 3, 5, 7, 11, 13, 17, 19, 23, and 29;
# of integers which are the sum of a positive multiple of 2 and an odd prime is 5: 9=7+2, 15=13+2, 21=19+2, 25=23+2 and 27=23+4;

Total: 14+9+5=28. You can see that we have all numbers from 1 to 30, not inclusive: 2, 3, 4, 5, 6, ...., 29.

Hope it's clear.

Hello Bunuel

If in such series, we get a number that repeats in both sets. Then do we have to count it once or twice?

For example:
How many positive integers less than 20 are multiple of 2 or a multiple of 3?

Multiple of 2: 2,4,6,8,10,12,14,16,18

Multiple of 3: 3,6,9,12,15,18

So do we have to count 6, 12, and 18 once or twice?

Total would be 15 or 12 ?

Thanks

How many positive integers less than 20 are multiple of 2 OR a multiple of 3?

Answer: 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18. So, total of 12 numbers.
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Director
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Posts: 630
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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19 Aug 2017, 02:59

Is it because of OR? If there was AND in place of, will the answer still be 12?
Math Expert
Joined: 02 Sep 2009
Posts: 62291
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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19 Aug 2017, 03:03
Shiv2016 wrote:

Is it because of OR? If there was AND in place of, will the answer still be 12?

How many positive integers less than 20 are multiple of both 2 and 3?

Answer: 6, 12, 18. Total of 3 numbers.
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WE: Engineering (Transportation)
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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19 Aug 2017, 03:29
multiple of 2 =14 which are even , 2,4,6,8,10........28
odd primes = 3,5,7,11,13,17,19,23,29 =9 numbers
odd prime and sum of multiple of 2 = 5,7,9,11,13,15,17,19,21,23,25,27,29
so total are 14+ (1) + 13 = 28
1 is used because 3 is only which is not there in 3rd list

total 28.
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Joined: 20 Jun 2017
Posts: 6
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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07 Oct 2017, 20:45
sir why can't we take 5=2+3and 31=2+29
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Joined: 11 Sep 2015
Posts: 4547
GMAT 1: 770 Q49 V46
How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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27 Oct 2018, 07:34
Top Contributor
enigma123 wrote:
How many positive integers less than 30 are either a multiple of 2, an odd prime number, or the sum of a positive multiple of 2 and an odd prime?

A. 29
B. 28
C. 27
D. 25
E. 23

Multiples of 2: 2, 4, 6, 8, 10, . . .26, 28

Sum of a positive multiple of 2 and an odd prime
3 is the smallest ODD prime
So, let's add multiples of 2 to 3.
We get: 3 + 2, 3 + 4, 3 + 6, 3 + 8, etc
Evaluate to get: 5, 7, 9, 11, . . . 27, 29

At this point, our list of numbers includes 2 as well as all integers from 4 to 29
All we're missing is 1 and 3

An odd prime number
3 is odd, so, now our list becomes: 2, 3, 4, 5, 6, . . . 27, 28, 29

So, the ONLY value that is NOT in the list is 1 (1 is NOT prime)

So, there are 28 numbers that meet the given conditions.

Cheers,
Brent
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Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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02 Nov 2019, 08:28
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Re: How many positive integers less than 30 are either a multiple of 2, an   [#permalink] 02 Nov 2019, 08:28

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