GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Feb 2019, 19:18

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT Prep Hour

February 20, 2019

February 20, 2019

08:00 PM EST

09:00 PM EST

Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST

February 21, 2019

February 21, 2019

10:00 PM PST

11:00 PM PST

Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

# How many positive integers less than 20 can be expressed as

Author Message
TAGS:

### Hide Tags

Intern
Joined: 12 Dec 2010
Posts: 13
Schools: Wharton, Columbia, Booth, NYU
How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

19 Dec 2010, 16:53
4
20
00:00

Difficulty:

85% (hard)

Question Stats:

57% (02:37) correct 43% (02:24) wrong based on 367 sessions

### HideShow timer Statistics

How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?

(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

_________________

Practice, More Practice, Still More Practice... THE only way to succeed!!!
+1Kudos, if this helps with your preparation. Good luck!

Director
Joined: 22 Mar 2011
Posts: 599
WE: Science (Education)
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

18 Aug 2012, 07:40
5
4
Mas[m]terGMAT12 wrote:
How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?
(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

The numbers must be of the form $$2a+3b,$$ where $$a$$ and $$b$$ are positive integers.
The smallest number is $$5 = 2*1 + 3*1.$$ Starting with $$5$$, we can get all the other numbers by adding either $$2$$ or $$3$$ to the already existing numbers on our list. Adding either $$2$$ or $$3$$ to $$2a+3b$$ will give another number of the same form.
So, after $$5$$, we get $$5+2=7, \,5+3=8, \,7+2=9, \,8+2=10,...$$ We will get all the numbers up to $$19$$ inclusive, except $$1,2,3,4,$$and $$6,$$ because once we have $$7$$ and $$8,$$ by adding $$2$$ all the time we can get any odd or even number.
We get a total of $$19-5=14$$ numbers.

Note: In fact, any integer $$n$$ greater than 6 has at least one representation of the form $$2a+3b.$$ If $$n$$ is odd, then $$n-3>2$$, so we can take $$b=1$$ and $$a=\frac{n-3}{2}.$$ If $$n$$ is even, being greater than $$6$$, $$n-6$$ is a positive multiple of $$2$$. Now we can take $$b=2$$ and $$a=\frac{n-6}{2}.$$
If the question would have been the same but for integers less than $$100$$, then the answer would be quite easy, $$99 - 5 = 94.$$
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

##### General Discussion
Retired Moderator
Joined: 02 Sep 2010
Posts: 763
Location: London
Re: More Number Properties Questions  [#permalink]

### Show Tags

19 Dec 2010, 23:43
1
MasterGMAT12 wrote:
What should be the approach to do the below question?

How many positive integers less than 20 can be expressed
as the sum of a positive multiple of 2 and a positive multiple
of 3?
(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

We are looking at the set {1,2,3,4,5,...,19}
So all numbers of the form 2+3k (where k>=1) can be considered {5,8,11,14,17} - set 1
Similarly 4+3k (k>=1) gets us {7,10,13,16,19} - set 2
6+3k (k>=1) gets us {9,12,15,18} - set 3
8+3k (k>=1) : already in set 1
10+3k (k>=1) : already in set 2
12+3k (k>=1) : already in set 3
14+3k (k>=1) : already in set 1
16+3k (k>=1) : already in set 2
18+3k (k>=1) : already in set 3

So the full list is {5,7,8,9,10,11,12,13,14,15,16,17,18,19} which is 14 numbers
_________________
Intern
Joined: 17 Jan 2012
Posts: 41
GMAT 1: 610 Q43 V31
Re: More Number Properties Questions  [#permalink]

### Show Tags

19 Feb 2012, 04:11
shrouded1 wrote:
MasterGMAT12 wrote:
What should be the approach to do the below question?

How many positive integers less than 20 can be expressed
as the sum of a positive multiple of 2 and a positive multiple
of 3?
(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

We are looking at the set {1,2,3,4,5,...,19}
So all numbers of the form 2+3k (where k>=1) can be considered {5,8,11,14,17} - set 1
Similarly 4+3k (k>=1) gets us {7,10,13,16,19} - set 2
6+3k (k>=1) gets us {9,12,15,18} - set 3
8+3k (k>=1) : already in set 1
10+3k (k>=1) : already in set 2
12+3k (k>=1) : already in set 3
14+3k (k>=1) : already in set 1
16+3k (k>=1) : already in set 2
18+3k (k>=1) : already in set 3

So the full list is {5,7,8,9,10,11,12,13,14,15,16,17,18,19} which is 14 numbers

Thanks for the Questions & Answer.

The mistake I did was that I constructed the equation as
Number = 2n+3n [i.e. 5,10,15] so my answer was "3" which was not there in the options. So I realized I m doing st wrong but I could not figure out until I saw the solution above.

The only problem was for me, above solution will take >2 min. Then I realized we can stop at 6+3k , because the # of numbers are already 14 ; the greatest answer option. Is there any other clue to look for?
Director
Joined: 27 May 2012
Posts: 683
Re: More Number Properties Questions  [#permalink]

### Show Tags

18 Aug 2012, 07:03
shrouded1 wrote:
MasterGMAT12 wrote:
What should be the approach to do the below question?

How many positive integers less than 20 can be expressed
as the sum of a positive multiple of 2 and a positive multiple
of 3?
(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

We are looking at the set {1,2,3,4,5,...,19}
So all numbers of the form 2+3k (where k>=1) can be considered {5,8,11,14,17} - set 1
Similarly 4+3k (k>=1) gets us {7,10,13,16,19} - set 2
6+3k (k>=1) gets us {9,12,15,18} - set 3
8+3k (k>=1) : already in set 1
10+3k (k>=1) : already in set 2
12+3k (k>=1) : already in set 3
14+3k (k>=1) : already in set 1
16+3k (k>=1) : already in set 2
18+3k (k>=1) : already in set 3

So the full list is {5,7,8,9,10,11,12,13,14,15,16,17,18,19} which is 14 numbers

although this solution is very helpful, but still I find the question a bit strange, without the solution it is almost impossible
to understand what the question is asking, I tried 2x + 3 and 2+3x as the number of elements, still no luck .

Can anybody make another attempt at this, thank you
_________________

- Stne

Manager
Joined: 23 May 2013
Posts: 102
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

01 Feb 2014, 20:13
we are looking for all positive numbers less than 20. That means we have 19 numbers.
Now, 1,2,3 and 4 can never be expressed as sum of 2 and 3. So we are left with 15 numbers.

By this time i already had spent around 3 min and had to take a shot, so i guessed it to 14.

Btw, i never came across an explanation where people would just guess the answers. I read that guessing is one of the skills that we need to master.
Anymore inputs to guessing will be welcomed
_________________

“Confidence comes not from always being right but from not fearing to be wrong.”

Manager
Joined: 22 Feb 2009
Posts: 163
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

06 Aug 2014, 20:35
3
2
MasterGMAT12 wrote:
How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?

(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

The number = 2a + 3b < 20

When a = 1, b = 1, 2, 3, 4, 5 -> 2a = 2; 3b = 3, 6, 9, 12, 15 -> the number = 5, 8, 11, 14, 17 --> 5 numbers
when a =2, b = 1,2,3,4,5 -> ....--> 5 numbers
when a =3, b = 1,2,3,4 --> ....--> 4 numbers

Total number is already 14. Look at the answer there is no number greater than 14 --> we dont need to try any more
_________________

.........................................................................
+1 Kudos please, if you like my post

Manager
Joined: 07 Feb 2015
Posts: 64
How many positive integers less than 20 can be expressed as the sum of  [#permalink]

### Show Tags

27 Jun 2015, 09:06
How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?

(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

Explanation: Positive multiples of 2 are even numbers; the relevant multiples of 3 are 3, 6, 9, 12, 15, and 18. No number smaller than 5 can be expressed as the sum of one and the other, as the smallest options are 2 and 3. Rather than going through every number between 5 and 19, look for patterns. There are 8 odd numbers between 5 and 19, inclusive, and each of them can be expressed as the sum of an even number and 3, so those 8 must be counted. The smallest even number that could be counted is 8 (2 + 6), and by the same reasoning, every even number between 8 and 18,inclusive, must be counted, adding 6 more to our total. That’s 6 + 8 = 14 total numbers, choice (A).
Manager
Joined: 07 Feb 2015
Posts: 64
Re: How many positive integers less than 20 can be expressed as the sum of  [#permalink]

### Show Tags

27 Jun 2015, 09:07
I don't understand this one. According to Bunuel here http://gmatclub.com/forum/is-0-zero-to-be-considered-as-a-multiple-of-every-number-104179.html, zero is a multiple of all numbers. Doesn't that mean the answer to this should be 19 (all are multiples except for one).
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

27 Jun 2015, 09:22
1
gmatser1 wrote:
How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?

(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

Explanation: Positive multiples of 2 are even numbers; the relevant multiples of 3 are 3, 6, 9, 12, 15, and 18. No number smaller than 5 can be expressed as the sum of one and the other, as the smallest options are 2 and 3. Rather than going through every number between 5 and 19, look for patterns. There are 8 odd numbers between 5 and 19, inclusive, and each of them can be expressed as the sum of an even number and 3, so those 8 must be counted. The smallest even number that could be counted is 8 (2 + 6), and by the same reasoning, every even number between 8 and 18,inclusive, must be counted, adding 6 more to our total. That’s 6 + 8 = 14 total numbers, choice (A).

_________________
Manhattan Prep Instructor
Joined: 22 Mar 2011
Posts: 1395
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

28 Jun 2015, 23:07
1
gmatser1, note that the problem specifies that we are dealing with positive multiples, so we don't need to consider 0. Otherwise, you would have a point. You'll find that little specifications like that (positive, not zero, integer, odd, even, etc.) are very important to take note of!
_________________

Dmitry Farber | Manhattan GMAT Instructor | New York

Manhattan GMAT Discount | Manhattan GMAT Course Reviews | View Instructor Profile |
Manhattan GMAT Reviews

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8883
Location: Pune, India
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

14 Feb 2016, 21:22
1
MasterGMAT12 wrote:
How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?

(A) 14
(B) 13
(C) 12
(D) 11
(E) 10

Responding to a pm:

I would do this question by enumerating and using pattern recognition.

Note that we need the number to be the sum of a positive multiple of 2 and a positive multiple of 3.
The first such number will be 5 (which is 2 + 3).
Now, every time we add one or more 2s and/or one or more 3s to 5, we will will one of our desired numbers.

$$5 +2 = 7$$

$$5+3 = 8$$

$$5 + 2*2 = 5 + 4 = 9$$

$$5 + 2 + 3 = 5 + 5 = 10$$

5 + 4 + 2 = 11

5 + 4 + 3 = 12

... Note that you will get all other numbers because the new base number is 5 + 4 = 9 now. You can add 2, 3, 4, 5 and 6. Thereafter, we can consider the new base to be 14 and then again add 2, 3, 4, 5, and 6 and so on...
So all numbers including and after 7 can be written in the form 2a + 3b.

In the first 19 positive integers, there are only 5 numbers (1, 2, 3, 4, 6) which you cannot express as 2a + 3b such that a and b are positive integers.
SO 14 numbers can be written as a sum of a positive multiple of 2 and a positive multiple of 3.

_________________

Karishma
Veritas Prep GMAT Instructor

Intern
Joined: 01 May 2015
Posts: 39
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

14 Feb 2016, 22:40
1
Positive multiple of 2 = 2,4,6,8,10,12,14,16,18,20
positive multiple of 3 = 3,6,9,12,15,18

So, various sums = 5, 8, 11, 14, 17, 7, 10, 13, 16, 19, 9, 12, 15, 18

This is a total of 14.
Non-Human User
Joined: 09 Sep 2013
Posts: 9855
Re: How many positive integers less than 20 can be expressed as  [#permalink]

### Show Tags

07 Oct 2018, 05:18
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: How many positive integers less than 20 can be expressed as   [#permalink] 07 Oct 2018, 05:18
Display posts from previous: Sort by