Problem Solving

Please, explain how to resolve this:

How many positive integers less than 20 are either a multiple of 2, an odd multiple of 9, or the sum of a positive multiple of 2 and a positive multiple of 9?

a)19
b)18
c)17
d)16
e)15

A = [2,4,6,8,10,12,14,16,18]

B = [9,18]

C = [13,15,17,19]

9+2+4=15

Fokin wrote:Please, explain how to resolve this:

How many positive integers less than 20 are either a multiple of 2, an odd multiple of 9, or the sum of a positive multiple of 2 and a positive multiple of 9?

a)19
b)18
c)17
d)16
e)15

Why C doesn't consist of 11 ?

outreach wrote:A = [2,4,6,8,10,12,14,16,18]

B = [9,18]

C = [13,15,17,19]

9+2+4=15

Fokin wrote:Please, explain how to resolve this:

How many positive integers less than 20 are either a multiple of 2, an odd multiple of 9, or the sum of a positive multiple of 2 and a positive multiple of 9?

a)19
b)18
c)17
d)16
e)15

e) 15

outreach wrote:A = [2,4,6,8,10,12,14,16,18]

B = [9,18]

C = [13,15,17,19]

9+2+4=15

Although the answer choice is correct, I think there are a couple of mistakes in the solution!

Here's my 2 cents worth!
In set B, you've considered 18, which is not an odd multiple of 9 and thus doesn't belong to this set. Hence, B = {9}
In set C we can also include 9+2 = 11 as it involves a positive multiple of 2 and a positive multiple of 9. Hence, C={11,13,15,17,19}
Therefore, total number of elements = 9 + 1 + 5 = 15
You got the answer right but the elements of the sets considered weren't accurate.

Thanks
Anirban

Hey guys,

Not to pick on outreach, but one other note on his solution is that he double-counts 18 in sets A and B, and that brings up a strategic point here:

When a question asks "either...or", you need to make sure you subtract out anything that you've double counted! Be very careful with either/or in quant questions (I've seen a lot of this variety that pertain to probability) as the GMAT will almost always give you something that can be double-counted and then screw up your result if you're not aware of it. (e.g. "what is the probability that a book will be either hardcover or a bestseller?" - any book that is a hardcover bestseller would be counted on both lists)

The way I looked at this problem was this - there are only 19 positive integers less than 20 (1-19 inclusive) and I can write a few digits per second, so I just wrote them down:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

Then, I started knocking them out as they fit a definition (eliminate the evens for A, 9 for B, and 11, 13, 15, 17, 19 for C). That way, those that left standing meet "none" of the categories, and I don't have to worry about double counting:

1
3
5
7

There are 19 total numbers and only 4 that meet none of the qualifications, so the remaining 15 meet at least one, and the answer is 15.

Two pretty big takeaways that I hope you walk away with from this one:

1) Beware of double-counting when you're asked to cross-reference categories!

2) Look for opportunities to subtract "none" from the total when it's easier than trying to determine how many items meet "at least one" of a set of characteristics.