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How many positive integers less than 30 are either a multiple of 2, an
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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
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Originally posted by enigma123 on 10 Feb 2012, 15:03.
Last edited by Bunuel on 08 Oct 2017, 03:15, edited 2 times in total.
Edited the question.




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Re: How many positive integers less than 30 are either a multiple of 2, an
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10 Feb 2012, 15:32
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 nonprime, 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 (282)/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 positive integers less than 30 are either a multiple of 2, an
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10 Feb 2012, 15:35
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 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
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05 Jun 2012, 19:59
Any odd number can be expressed as 2k+1 or 2k+(32) or 2(K1)+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 =291.



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Re: How many positive integers less than 30 are either a multiple of 2, an
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05 Jun 2012, 23:26
asax wrote: Any odd number can be expressed as 2k+1 or 2k+(32) or 2(K1)+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 =291. Definitely very clever. I spent 2 minutes going the long way until I realized that.



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Re: How many positive integers less than 30 are either a multiple of 2, an
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21 Aug 2012, 15:15
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 nonprime, 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 (282)/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 positive integers less than 30 are either a multiple of 2, an
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22 Aug 2012, 00:22
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 nonprime, 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 (282)/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 positive integers less than 30 are either a multiple of 2, an
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24 Aug 2012, 12:27
Bunuel wrote: alphabeta1234 wrote: 30 sec approach: Any odd nonprime, 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 (282)/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+CABACBCABC+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 multiple of 2, an
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22 Sep 2012, 13:56
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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Re: 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
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26 Jun 2013, 08:26
enigma123 wrote: 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 Qquestion: 0<x<30 so, 1<=x<=29 leave x=1 alone for a while, and consider everything else i.e. 2<=x<=29 integer either multiple of 2 that will be almost half the no's (14) odd prime no, and sum of a positive multiple of 2 and an odd prime => Rest everything else has to be either a prime no or the sum of some multiple of 2(Those 14 no we got earlier)and a odd no only for x=1, it is neither even, nor prime and definitely not the sum. Thus ans = total no's  1 = 29  1 = 28 Ans: B
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Re: How many positive integers less than 30 are either a multiple of 2, an
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04 Oct 2013, 22:09
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 nonprime, 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 (282)/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. I did not understand the last condition ? sum of a positive multiple of 2 and an odd prime ? it can be possible: 7=5+2 ???



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Re: How many positive integers less than 30 are either a multiple of 2, an
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05 Oct 2013, 04:12
sunny3011 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 nonprime, 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 (282)/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. I did not understand the last condition ? sum of a positive multiple of 2 and an odd prime ? it can be possible: 7=5+2 ??? 2 is not an odd prime. But 7 CAN be written as the sum of a positive multiple of 2 and an odd prime: 7 = 4 + 3.
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Re: How many positive integers less than 30 are either a multiple of 2, an
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16 Oct 2013, 23:26
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 nonprime, 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 (282)/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. In this # of integers which are the sum of a positive multiple of 2 and an odd prime ,.. why didnt we count 7=5+2 and 13=11+2,19=13+4 .. ??? these all are Sum of multiple of 2 and odd primes. ????



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Re: How many positive integers less than 30 are either a multiple of 2, an
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17 Oct 2013, 02:09
ishdeep18 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 nonprime, 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 (282)/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. In this # of integers which are the sum of a positive multiple of 2 and an odd prime ,.. why didnt we count 7=5+2 and 13=11+2,19=13+4 .. ??? these all are Sum of multiple of 2 and odd primes. ???? Because 7, 13, and 19 (all primes) are included in the second set (dd primes).
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Re: How many positive integers less than 30 are either a multiple of 2, an
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23 Dec 2013, 23:03
What are the actual 2 numbers that answer this question? I know 1 is one of them, but I can't think of the other one...I used to think it was 0 but technically 0 is neither positive nor negative...



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Re: 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
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24 Dec 2013, 10:48
Bunuel wrote: catalysis wrote: What are the actual 2 numbers that answer this question? I know 1 is one of them, but I can't think of the other one...I used to think it was 0 but technically 0 is neither positive nor negative... I think you misinterpreted the question. It asks: " how many positive integers less than 30 are ..." Hi Bunuel  Sorry, I think I misworded my original question. I know the answer is 28, which means 28 numbers less than 30 meet the constraints given. However, I was just curious which values are the numbers that do NOT meet the constraints. However, I have kind of answered my own question because now I realize that there are only 29 integers to choose from (129 inclusive), not 30 like I had originally thought, because 0 is not a positive integer and 30 cannot be included because the question asks for numbers less than 30. Therefore, it makes sense that 1 is the only integer that does not meet the constraints and I should not be looking for a second number. (29 possible integers  1 integer that does not meet the constraints = 28 integers that meet the constraints, just like the answer says) Hope this makes sense...



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Re: How many positive integers less than 30 are either a multiple of 2, an
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07 Nov 2016, 04:41
enigma123 wrote: 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 Let’s use PIE principle to solve this question. \(XUYUZ = X + Y + Z  X⋂Y  X⋂Z  Y⋂Z + X⋂Y⋂Z\) We have: \(X\)  “multiples of 2” – even numbers between 1 and 29 = 14 \(Y\)  “odd prime numbers” – 3, 5, 7, 11, 13, 17, 19, 23, 29 = 9 \(Z\) “sum of positive multiple of 2 and odd prime” (2a+p), where p is odd prime. This function generates all odd numbers except 1 and 3. 1 – because we have positive multiple of 2 (a≠0), and 3 – because we need to add prime number and in order to generate 3 we need to add 1, which is not prime. So we have total # of odd integers in the range minus 1 and 3: 15 – 2 = 13. \(X⋂Y\) = 0 = because the number cannot be simultaneously even and odd prime \(X⋂Z\) = 8  number is simultaneously prime and generated by the function 2a+p, and we know that this function cannot generate prime 3. So we have 91 = 8 \(X⋂Z\) = 0  can’t be simultaneously even and odd. \(X⋂Y⋂Z\) = 0 – same logic as in previous case. The resultant # is = 14 + 9 + 13 – 8 = 28




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