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

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

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New post 19 Aug 2017, 02:06
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.

Answer: B.

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

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

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New post 19 Aug 2017, 03:31
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.

Answer: B.

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

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New post 19 Aug 2017, 03:59
Thank you Bunuel for your reply.

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

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New post 19 Aug 2017, 04:03

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

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New post 19 Aug 2017, 04:29
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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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Re: How many positive integers less than 30 are either a multiple of 2, an [#permalink]

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New post 07 Oct 2017, 21:45
sir why can't we take 5=2+3and 31=2+29

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Re: How many positive integers less than 30 are either a multiple of 2, an   [#permalink] 07 Oct 2017, 21:45

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