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How many positive integers n have the property that both 3n and n/3 ar

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How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 21 Sep 2019, 13:53
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Question Stats:

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How many positive integers n have the property that both 3n and n/3 are 4-digit integers?

A. 111
B. 112
C. 333
D. 334
E. 1,134

PS04851.01
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 21 Sep 2019, 14:00
3
2
Minimum 4 digit number is 1000
Maximum 4 digit number is 9999
max=3n=9999
nmax=3333
min=n3=1000
nmin=3000
Keep in mind that n must be divisible by 3. IMO: the answer would be:
3333−30003+1=112
B

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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 23 Sep 2019, 04:34
3
1
4 digit nos
for starts min value 3n at n= 334; 1002 and max at n= 3333 i.e 9999
now the limit given is positive integers n have the property that both 3n and n/3 are 4-digit integers
so n/3 ; 4 digit starts at n=3000 and ends at n=9999
the common range for the given condition is n= 3333-3000; 333 integers i.e values would be 333/3+1 ; 112
IMO b



gmatt1476 wrote:
How many positive integers n have the property that both 3n and n/3 are 4-digit integers?

A. 111
B. 112
C. 333
D. 334
E. 1,134

PS04851.01
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 28 Sep 2019, 07:29
1
Archit3110 wrote:
4 digit nos
for starts min value 3n at n= 334; 1002 and max at n= 3333 i.e 9999
now the limit given is positive integers n have the property that both 3n and n/3 are 4-digit integers
so n/3 ; 4 digit starts at n=3000 and ends at n=9999
the common range for the given condition is n= 3333-3000; 333 integers i.e values would be 333/3+1 ; 112
IMO b



gmatt1476 wrote:
How many positive integers n have the property that both 3n and n/3 are 4-digit integers?

A. 111
B. 112
C. 333
D. 334
E. 1,134

PS04851.01



Why do we add the 1 in denominator for final answer i.e 333/3+1 ??
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 29 Sep 2019, 08:27
3
1

Solution



To find

    • The number of integers such that both 3n and n/3 are 4-digit integers.

Approach and Working out

    • 1000 < =3n <= 9999
      o 1000/3 <= n < = 9999/3
      o 333.33 < = n < 3333-----(1)
    • 1000 < =n/3 <= 9999
      o 3000 < = n < 9999*3---------(2)

    • Combining both 1 and 2, we get
      o 3000 <= n < 3333
      o As n/3 is an integer, n has to be a multiple of 3.
      o (3333-3000)/3 +1 = 112
Hence, n can have 112 values.

Thus, option B is the correct answer.

Correct Answer: Option B
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 29 Sep 2019, 09:20
Vinayak013 wrote:
Archit3110 wrote:
4 digit nos
for starts min value 3n at n= 334; 1002 and max at n= 3333 i.e 9999
now the limit given is positive integers n have the property that both 3n and n/3 are 4-digit integers
so n/3 ; 4 digit starts at n=3000 and ends at n=9999
the common range for the given condition is n= 3333-3000; 333 integers i.e values would be 333/3+1 ; 112
IMO b



gmatt1476 wrote:
How many positive integers n have the property that both 3n and n/3 are 4-digit integers?

A. 111
B. 112
C. 333
D. 334
E. 1,134

PS04851.01



Why do we add the 1 in denominator for final answer i.e 333/3+1 ??


We add +1 because when we subtract two numbers, let's say 3 and 1. The result is the gap between them. 3-1 = 2 (Gap between 3 and 1)

So, if we want to include all the three numbers, we need to add 1 to it.

I hope this helps you.
Regards,
Ashutosh
e-GMAT Quant Expert
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 29 Sep 2019, 11:06
Still not clear can you Please elaborate?
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 29 Sep 2019, 14:32
1
Hey @Vinayaka,

Okay, let's answer a few of my question and i am sure you will understand what I am trying to say.

How many numbers are there from 1 to 10, both inclusive?
Is the answer 10?
Is the answer equal to 10-1?
Is the answer equal to (10-1) +1?

You will find that answer is equal to (10-1)+1 and this is because the difference between two numbers gives the gap between two numbers and, total consecutive number that forms the gaps is always 1 greater than the gap.

So, between 1 and 10, there are 9 gaps and it formed by 10 consecutive numbers.

I hope this helps you.
Regards,
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 30 Sep 2019, 06:38
1
n= x/3 = y*3 where x and y are 4 digit integers.

Therefore, x = 9*y

Min y = 1000, Corresponding Min x= 9000
Max x= 9999, Corresponding Max y= 1111

Possible values of n= (1111-1000) + 1 = 112 OR (9999-9000)/9 + 1 = 112

Ans B
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 04 Oct 2019, 05:09
EgmatQuantExpert wrote:
Hey @Vinayaka,

Okay, let's answer a few of my question and i am sure you will understand what I am trying to say.

How many numbers are there from 1 to 10, both inclusive?
Is the answer 10?
Is the answer equal to 10-1?
Is the answer equal to (10-1) +1?

You will find that answer is equal to (10-1)+1 and this is because the difference between two numbers gives the gap between two numbers and, total consecutive number that forms the gaps is always 1 greater than the gap.

So, between 1 and 10, there are 9 gaps and it formed by 10 consecutive numbers.

I hope this helps you.
Regards,




Got it , Thanks a ton!
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 17 Oct 2019, 15:16
EgmatQuantExpert wrote:

Solution



To find

    • The number of integers such that both 3n and n/3 are 4-digit integers.

Approach and Working out

    • 1000 < =3n <= 9999
      o 1000/3 <= n < = 9999/3
      o 333.33 < = n < 3333-----(1)
    • 1000 < =n/3 <= 9999
      o 3000 < = n < 9999*3---------(2)

    • Combining both 1 and 2, we get
      o 3000 <= n < 3333
      o As n/3 is an integer, n has to be a multiple of 3.
      o (3333-3000)/3 +1 = 112
Hence, n can have 112 values.

Thus, option B is the correct answer.

Correct Answer: Option B


If 3000 <= n < 3333 then why isn't the answer 334? There are 334 integers in this range..
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Re: How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 18 Oct 2019, 17:45
jamalabdullah100 wrote:
EgmatQuantExpert wrote:

Solution



To find

    • The number of integers such that both 3n and n/3 are 4-digit integers.

Approach and Working out

    • 1000 < =3n <= 9999
      o 1000/3 <= n < = 9999/3
      o 333.33 < = n < 3333-----(1)
    • 1000 < =n/3 <= 9999
      o 3000 < = n < 9999*3---------(2)

    • Combining both 1 and 2, we get
      o 3000 <= n < 3333
      o As n/3 is an integer, n has to be a multiple of 3.
      o (3333-3000)/3 +1 = 112
Hence, n can have 112 values.

Thus, option B is the correct answer.

Correct Answer: Option B


If 3000 <= n < 3333 then why isn't the answer 334? There are 334 integers in this range..


We obtained the more defined range be looking at potential values for 3n and n/3.

Because n/3 must be an integer we are only concerned with integers divisible by 3, so we only count those integers.

We count integers divisible by a specific integer such as 3 by (upper range - lower range)/multiple we try to count + 1
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How many positive integers n have the property that both 3n and n/3 ar  [#permalink]

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New post 28 Oct 2019, 15:15
EgmatQuantExpert wrote:

Solution



To find

    • The number of integers such that both 3n and n/3 are 4-digit integers.

Approach and Working out

    • 1000 < =3n <= 9999
      o 1000/3 <= n < = 9999/3
      o 333.33 < = n < 3333-----(1)
    • 1000 < =n/3 <= 9999
      o 3000 < = n < 9999*3---------(2)

    • Combining both 1 and 2, we get
      o 3000 <= n < 3333
      o As n/3 is an integer, n has to be a multiple of 3.
      o (3333-3000)/3 +1 = 112
Hence, n can have 112 values.

Thus, option B is the correct answer.

Correct Answer: Option B


Hey egmat
EgmatQuantExpert

You missed the '=' sign here.
I think it should be
333.33 < = n <= 3333-----(1)[/list]

And you missed the '=' sign here also.
IMO it should be
3000 < = n <= 9999*3


Again you missed it here.
It should be ,

Combining both 1 and 2, we get
[list]o 3000 <= n < = 3333


Please check once. :)
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How many positive integers n have the property that both 3n and n/3 ar   [#permalink] 28 Oct 2019, 15:15
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