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

Definitely very clever. I spent 2 minutes going the long way until I realized that.

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?

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.

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).

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 ..."

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

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.

Thank you Bunuel for your reply.

Is it because of OR? If there was AND in place of, will the answer still be 12?

How many positive integers less than 20 are multiple of both 2 and 3?

Answer: 6, 12, 18. Total of 3 numbers.

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.

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

GMATPrepNow

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

Thanks,
K