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

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Question Stats: 38% (02:34) correct 62% (02:40) wrong based on 737 sessions

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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, 16:03.
Last edited by Bunuel on 08 Oct 2017, 04:15, edited 2 times in total.
Edited the question.
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Posts: 58449
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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15
28
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.
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Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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

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enigma123 wrote:
Many thanks Bunuel - you mean to say answer is B. I take it's a typo at your end Yes, 28 is answer B.
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GMAT 1: 620 Q46 V30 GMAT 2: 720 Q50 V38 Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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2
1
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: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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

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2
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.

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 58449
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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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.

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

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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.

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 multiple of 2, an  [#permalink]

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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  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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

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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  [#permalink]

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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.

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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Posts: 58449
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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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 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.

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  [#permalink]

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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.

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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Posts: 58449
Re: How many positive integers less than 30 are either a multiple of 2, an  [#permalink]

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1
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 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.

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  [#permalink]

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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  [#permalink]

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

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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 (1-29 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  [#permalink]

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2
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 9-1 = 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 Re: How many positive integers less than 30 are either a multiple of 2, an   [#permalink] 07 Nov 2016, 05:41

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