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

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

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19 Dec 2010, 17:53
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75% (hard)

Question Stats:

57% (02:02) correct 43% (02:04) wrong based on 350 sessions

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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

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Director
Joined: 22 Mar 2011
Posts: 601
WE: Science (Education)
Re: How many positive integers less than 20 can be expressed as  [#permalink]

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18 Aug 2012, 08:40
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, $$99 - 5 = 94.$$
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Re: More Number Properties Questions  [#permalink]

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20 Dec 2010, 00:43
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
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Re: More Number Properties Questions  [#permalink]

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19 Feb 2012, 05:11
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: 586
Re: More Number Properties Questions  [#permalink]

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18 Aug 2012, 08:03
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
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Re: How many positive integers less than 20 can be expressed as  [#permalink]

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01 Feb 2014, 21:13
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
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Posts: 175
Re: How many positive integers less than 20 can be expressed as  [#permalink]

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06 Aug 2014, 21:35
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
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How many positive integers less than 20 can be expressed as the sum of  [#permalink]

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27 Jun 2015, 10:06
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).
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Posts: 67
Re: How many positive integers less than 20 can be expressed as the sum of  [#permalink]

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27 Jun 2015, 10:07
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).
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Joined: 02 Sep 2009
Posts: 50007
Re: How many positive integers less than 20 can be expressed as  [#permalink]

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27 Jun 2015, 10:22
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).

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Re: How many positive integers less than 20 can be expressed as  [#permalink]

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29 Jun 2015, 00:07
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!
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Re: How many positive integers less than 20 can be expressed as  [#permalink]

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14 Feb 2016, 22:22
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.

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Re: How many positive integers less than 20 can be expressed as  [#permalink]

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14 Feb 2016, 23:40
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.
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Re: How many positive integers less than 20 can be expressed as  [#permalink]

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