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How many positive integers less than 20 can be expressed as

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How many positive integers less than 20 can be expressed as [#permalink] New post 19 Dec 2010, 16:53
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Question Stats:

57% (02:57) correct 43% (01:37) wrong based on 121 sessions
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
[Reveal] Spoiler: OA

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Re: More Number Properties Questions [#permalink] New post 19 Dec 2010, 23:43
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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] New post 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?
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Re: More Number Properties Questions [#permalink] New post 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
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Re: How many positive integers less than 20 can be expressed as [#permalink] New post 18 Aug 2012, 07:40
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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.

Answer A

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: How many positive integers less than 20 can be expressed as [#permalink] New post 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
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Re: How many positive integers less than 20 can be expressed as [#permalink] New post 06 Aug 2014, 20:35
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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
Answer must be A
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Re: How many positive integers less than 20 can be expressed as   [#permalink] 06 Aug 2014, 20:35
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