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How many positive integers less than 20 are ?

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Junior | Next Rank: 30 Posts
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by Fokin » Tue Aug 10, 2010 7:52 am
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
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by outreach » Tue Aug 10, 2010 8:07 am
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
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by Fokin » Tue Aug 10, 2010 8:14 am
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
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by commonsense27 » Tue Aug 10, 2010 8:18 am
e) 15
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by anirban_lax » Tue Aug 10, 2010 8:37 am
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
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by Brian@VeritasPrep » Tue Aug 10, 2010 11:35 am
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.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep

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