# Thread: Sums of different multiples

1. ## Sums of different multiples

I am currently studying for the GMAT, and I have run into a problem that confuses me because the correct answer explanation is somewhat vague.

How many positive integers less than 28 are prime numbers, odd multiples of 5, or the sum of a positive multiple of 2 and a positive multiple of 4?

A 27
B 25
C 24
D 22
E 20

The correct answer is D, 22

This is how I am solving the problem.

Step 1: Write out all of the numbers and circle all of the prime numbers. This leaves me with 1, 3, 5, 7, 11, 13, 17, 19, 23, and 27 (10 terms)

Step 2: Include all odd multiples of 5 that are not already prime, which was 15 and 25 (12 terms so far)

Step 3:???

My book's explanation says that a positive sum of a multiple of 2 and 4 just means to include all even numbers after 4. However, I do not understand the logic. For example, how does 6 contain a multiple of 4 AND a multiple of 2? etc.

2. ## Re: Sums of different multiples

6 = 1x4 + 1x2...

3. ## Re: Sums of different multiples

Originally Posted by Prove It
6 = 1x4 + 1x2...
Ah I see now. I was thinking of it as divisors instead of factors.

4. ## Re: Sums of different multiples

Originally Posted by KingNathan
Step 1: Write out all of the numbers and circle all of the prime numbers.
This leaves me with 1, 3, 5, 7, 11, 13, 17, 19, 23, and 27 (10 terms)
You mean 2,3,5, ....
OK?

5. ## Re: Sums of different multiples

Hello, KingNathan!

How many positive integers less than 28 are prime numbers, odd multiples of 5,
or the sum of a positive multiple of 2 and a positive multiple of 4?

. . $\displaystyle (A)\;27 \qquad (B)\;25\qquad (C)\;24 \qquad (D)\;22 \qquad (E)\;20$

The correct answer is: (D) 22

The numbers are less than or equal to 27.

Primes: .{2, 3, 5, 7, 11, 13, 17, 19, 23}

Odd multiples of 5: .{5, 15, 25}

Sum of $\displaystyle 2a$ and $\displaystyle 4b\!:$ .{6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28}

The union of these sets has 22 elements.
. . (Don't count the "5" twice.)