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Bunuel
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What is the value of a positive integer n if 2, 3 and 5 are the only 22 Mar 2023, 05:30
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C
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75% (hard)

Question Stats:

42% (02:04) correct 58% (01:33) wrong based on 57 sessions

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What is the value of a positive integer n if 2, 3 and 5 are the only prime factors of n ?

(1) n > 100
(2) n has exactly 12 factors.
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C
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Rabab36
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What is the value of a positive integer n if 2, 3 and 5 are the only 22 Mar 2023, 12:38
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statement 1

n can be 120,150,180

not sufficent


statement 2

only 30 is having 12 factors with 2, 3 and 5 as prime factors
suffcient


option B
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What is the value of a positive integer n if 2, 3 and 5 are the only 23 Mar 2023, 06:34
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Rephrase the statement - We have to find a multiple of 30..

Statement 1 .. n>100

Multiples of 30 greater than 100 = 120, 150, 180, 210 etc.

Not sufficient

Statement 2

Has exactly 12 factors

Insufficient ..

Combining A and B WE GET 150 as the number which has 12 factors .
150 = 3^1 * 2^1 * 5^2 total factors .. 2*2*3. = 12

120 , 180 etc have 18 factors

Hence C is sufficient
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What is the value of a positive integer n if 2, 3 and 5 are the only 15 May 2023, 20:28
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Bunuel
What is the value of a positive integer n if 2, 3 and 5 are the only prime factors of n ?

(1) n > 100
(2) n has exactly 12 factors.
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Statement 1

\(2^2 . 3^2 . 5^2 = 900\) which is greater than 100
\(2^2 . 3^3 . 5^ 2 = 2700\) which is also greater than 100

Not Sufficient

Statement 2

If the number can be written in the form of - \(2^a . 3^b . 5^c\)
Then we know : (a+1)(b+1)(c+1) = 12

Possible values-
a=1, b=1, c=2
a=1, b=2, c=1
a=2, b=1, c=1

We have 3 possible scenarios
\(2^1 . 3^1 . 5^2 = 150\)
\(2^1 . 3^2 . 5^1 = 90\)
\(2^2 . 3^1 . 5^1 = 60\)

Not Sufficient

Combining together

Only scenario with a=1, b=1, c=2 has value of n>100
Therefore sufficient.
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What is the value of a positive integer n if 2, 3 and 5 are the only 15 May 2023, 23:42
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Bunuel
What is the value of a positive integer n if 2, 3 and 5 are the only prime factors of n ?

(1) n > 100
(2) n has exactly 12 factors.
Show more
Solution:
Pre Analysis:
  • It is given that \(n=2^a\times 3^b\times 5^c\) where a, b and c are positive integers
  • We are also asked the value of \(n\)

Statement 1: n > 100
  • If \(a=1, b=1\) and \(c=2\), then \(n=2^1\times 3^1\times 5^2=150\) which is greater than 100
  • And if \(a=2, b=1\) and \(c=2\), then \(n=2^2\times 3^1\times 5^2=300\) which is also greater than 100
  • Thus, statement 1 alone is not sufficient and we can eliminate options A and D

Statement 2: n has exactly 12 factors
  • We know \(12=2\times 2\times 3\)
  • So, \(a+1, b+1\) and \(c+1\) has to be \(2, 2\) and \(3\) in any order
  • There are three possible scenarios:
    • \(n=2^1\times 3^1\times 5^2=150\)
    • \(n=2^1\times 3^2\times 5^1=90\) and
    • \(n=2^2\times 3^1\times 5^1=60\)
  • Thus, statement 2 alone is also not sufficient

Combining:
  • From statement 1,we get that \(n>100\)
  • From statement 2, we get that \(n=150,90\) or \(60\)
  • Thus, we can confirm \(n=150\)

Hence the right answer is Option C
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