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How many positive integers less than 30 are either a [#permalink]
 10 Feb 2012, 16:03
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How many positive integers less than 30 are either a multiple of 2, an odd prime number, of the sum of a positive multiple of 2 and an odd prime? A. 29 B. 28 C. 27 D. 25 E. 23
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Re: How many +ve integers? [#permalink]
10 Feb 2012, 16:32
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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.
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Re: How many +ve integers? [#permalink]
10 Feb 2012, 16:35
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Many thanks Bunuel - you mean to say answer is B. I take it's a typo at your end 
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Re: How many +ve integers? [#permalink]
10 Feb 2012, 16:36
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How many positive integers less than 30 are either a multiple of 2, an odd prime number, of the sum of a positive multiple of 2 and an odd prime? (A) 29 (B) 28 (C) 27 (D) 25 (E) 23 is there any shortcut method to solve this type of problem??
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Re: short cut method [#permalink]
05 Jun 2012, 20:59
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Any odd number can be expressed as 2k+1 or 2k+(3-2) or 2(K-1)+3. Thus, with the prime number 3, we can express all the odd numbers. Since, 1 i is the only number that cannot be expressed, answer is numbers <30 =29-1.
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Re: short cut method [#permalink]
06 Jun 2012, 00:26
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asax wrote: Any odd number can be expressed as 2k+1 or 2k+(3-2) or 2(K-1)+3. Thus, with the prime number 3, we can express all the odd numbers.
Since, 1 i is the only number that cannot be expressed, answer is numbers <30 =29-1. Definitely very clever. I spent 2 minutes going the long way until I realized that.
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Re: How many +ve integers? [#permalink]
21 Aug 2012, 16:15
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. Hey Bunuel, How can this be the entire list? # 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; Shouldnt be: 2(1)+3<30 2(1)+5<30 2(1)+7<30 2(1)+11<30 .... 2(1)+23<30 Now 2(2)+3<30 2(2)+5<30 2(2)+7<30 2(2)+11<30 .... 2(2)+23<30 etc Your list didn't include all those? What am I missing?
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Re: How many +ve integers? [#permalink]
22 Aug 2012, 01:22
alphabeta1234 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. Hey Bunuel, How can this be the entire list? # 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; Shouldnt be: 2(1)+3<30 2(1)+5<30 2(1)+7<30 2(1)+11<30 .... 2(1)+23<30 Now 2(2)+3<30 2(2)+5<30 2(2)+7<30 2(2)+11<30 .... 2(2)+23<30 etc Your list didn't include all those? What am I missing? First of all we are asked about the number of positive integers less than 30, which are a multiple of 2 OR an odd prime number OR the sum of a positive multiple of 2 and an odd prime. Next, EACH numbers from 1 to 30, not inclusive is a multiple of 2 OR an odd prime number OR the sum of a positive multiple of 2 and an odd prime. So, the list is 2, 3, 4, 5, ..., 29 (total of 28 numbers). So, which number is not included in the list?
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Re: How many +ve integers? [#permalink]
24 Aug 2012, 13:27
Bunuel wrote: alphabeta1234 wrote:
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.
Hey Bunuel, How can this be the entire list? # 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; Shouldnt be: 2(1)+3<30 2(1)+5<30 2(1)+7<30 2(1)+11<30 .... 2(1)+23<30 Now 2(2)+3<30 2(2)+5<30 2(2)+7<30 2(2)+11<30 .... 2(2)+23<30 etc Your list didn't include all those? What am I missing? Bunuel's Response: First of all we are asked about the number of positive integers less than 30, which are a multiple of 2 OR an odd prime number OR the sum of a positive multiple of 2 and an odd prime. Next, EACH numbers from 1 to 30, not inclusive is a multiple of 2 OR an odd prime number OR the sum of a positive multiple of 2 and an odd prime. So, the list is 2, 3, 4, 5, ..., 29 (total of 28 numbers). So, which number is not included in the list?[/quote] Hey Bunuel, Thanks for pointing out my mistake the same numbers that are generated by 2K+odd prime are also included in the same list as the odd primes. In other words A=# of even numbers between 1 and 29, inclusive B=# of odd primes between 1 and 29, inclusive C=# of 2K+odd_prime, between 1 and 29, inclusive AUBUC=A+B+C-AB-AC-BC-ABC+N AB=0, since there are no numbers both even and odd primes between 1 and 29, inclusive AC=0, since there are no numbers both even and 2K+odd_prime(=odd) between 1 and 29, inclusive ABC=0 since no numbers are even, and odd prime and a 2K+odd_prime and N=1, since only 1 fits the criteria of being niether an even number, neither an odd prime, and neither a 2K+odd_prime My question I guess is for BC, numbers both an odd prime and 2K+odd_prime. Is there a way to tell, without actually listing out all the numbers that meet this condition and checking ? Thank you!
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Re: How many positive integers less than 30 are either a [#permalink]
22 Sep 2012, 14:56
I can't believe that what made this problem difficult was a "typo error" in the question statement!!!!
Instead of "... number, of the sum of a positive multiple..." is "... number, OR the sum of a positive...
Thank you for clarifying!!! =)
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How many positive integers less than 30 are either [#permalink]
04 Mar 2013, 11:24
How many positive integers less than 30 are either a multiple of 2, an odd prime number, of the sum of a positive multiple of 2 and an odd prime?(A) 29 (B) 28 (C) 27 (D) 25 (E) 23
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Re: How many positive integers less than 30 are either [#permalink]
04 Mar 2013, 11:27
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Re: How many positive integers less than 30 are either
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