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How many positive integers less than 30 are either a multiple of 2, an Updated on: 08 Oct 2017, 04:15
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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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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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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B
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How many positive integers less than 30 are either a multiple of 2, an 10 Feb 2012, 16:32
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enigma123
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?
Show more

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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How many positive integers less than 30 are either a multiple of 2, an 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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How many positive integers less than 30 are either a multiple of 2, an 06 Jun 2012, 00:26
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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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Definitely very clever. I spent 2 minutes going the long way until I realized that.
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How many positive integers less than 30 are either a multiple of 2, an 21 Aug 2012, 16:15
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Bunuel
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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

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.
Show more

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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How many positive integers less than 30 are either a multiple of 2, an 22 Aug 2012, 01:22
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Bunuel
enigma123
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?
Show more

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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How many positive integers less than 30 are either a multiple of 2, an 04 Oct 2013, 23:09
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Bunuel
enigma123
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.
Show more


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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How many positive integers less than 30 are either a multiple of 2, an 05 Oct 2013, 05:12
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sunny3011
Bunuel
enigma123
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.


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 ???
Show more

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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How many positive integers less than 30 are either a multiple of 2, an 17 Oct 2013, 00:26
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Bunuel
enigma123
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.
Show more

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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How many positive integers less than 30 are either a multiple of 2, an 17 Oct 2013, 03:09
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Bunuel
enigma123
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.

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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Because 7, 13, and 19 (all primes) are included in the second set (dd primes).
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How many positive integers less than 30 are either a multiple of 2, an 24 Dec 2013, 00:03
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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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How many positive integers less than 30 are either a multiple of 2, an 24 Dec 2013, 01:33
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catalysis
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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I think you misinterpreted the question. It asks: "how many positive integers less than 30 are ..."
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How many positive integers less than 30 are either a multiple of 2, an 07 Nov 2016, 05:41
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enigma123
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
Show more

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
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How many positive integers less than 30 are either a multiple of 2, an 19 Aug 2017, 02:06
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Bunuel
enigma123
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.
Show more


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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How many positive integers less than 30 are either a multiple of 2, an 19 Aug 2017, 03:31
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Bunuel
enigma123
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
Show more

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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How many positive integers less than 30 are either a multiple of 2, an 19 Aug 2017, 03:59
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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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How many positive integers less than 30 are either a multiple of 2, an 19 Aug 2017, 04:03
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Shiv2016
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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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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How many positive integers less than 30 are either a multiple of 2, an 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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How many positive integers less than 30 are either a multiple of 2, an 27 Oct 2018, 08:34
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enigma123
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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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.

Answer: B

Cheers,
Brent
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How many positive integers less than 30 are either a multiple of 2, an 12 Apr 2020, 14:38
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GMATPrepNow

Why did you choose 3 to add to multiples of 2 ?

Thanks,
K
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