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Bunuel
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If the positive integer N has only three prime factors 2, 3, and 5 30 Jan 2023, 12:15
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55% (hard)

Question Stats:

58% (01:43) correct 42% (01:29) wrong based on 78 sessions

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If the positive integer N has only three prime factors 2, 3, and 5, what is the value of N ?

(1) N has a total of 12 factors
(2) N > 100
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C
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gmatophobia
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If the positive integer N has only three prime factors 2, 3, and 5 30 Jan 2023, 12:57
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Bunuel
If the positive integer N has only three prime factors 2, 3, and 5, what is the value of N ?

(1) N has a total of 12 factors
(2) N > 100
Show more

Given

N has three prime factors

Statement 1

(1) N has a total of 12 factors

Possible values of N = \(2^2 * 3 * 5\) , \(2 * 3^2 * 5\) or \(2 * 3 * 5^2\)

As we can have multiple values of N, the statement is not sufficient and we can eliminate A and D.

Statement 2

(2)N > 100

Insufficient, eliminate B.

Combined

The statements combined we can eliminate the possibility of N = \(2^2 * 3 * 5\) and N = \(2 * 3^2 * 5\) as both of these numbers are less than 100.

Hence N = \(2 * 3 * 5^2\)

Sufficient.

Option C
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If the positive integer N has only three prime factors 2, 3, and 5 04 Jun 2024, 04:26
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gmatophobia

Bunuel
If the positive integer N has only three prime factors 2, 3, and 5, what is the value of N ?

(1) N has a total of 12 factors
(2) N > 100
Given

N has three prime factors

Statement 1

(1) N has a total of 12 factors

Possible values of N = \(2^2 * 3 * 5\) , \(2 * 3^2 * 5\) or \(2 * 3 * 5^2\)

As we can have multiple values of N, the statement is not sufficient and we can eliminate A and D.

Statement 2

(2)N > 100

Insufficient, eliminate B.

Combined

The statements combined we can eliminate the possibility of N = \(2^2 * 3 * 5\) and N = \(2 * 3^2 * 5\) as both of these numbers are less than 100.

Hence N = \(2 * 3 * 5^2\)

Sufficient.

Option C
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­Hi would you be able to give an explanation/rule that gives the result of (1) N has a total of 12 factors - Possible values of N = \(2^2 * 3 * 5\) , \(2 * 3^2 * 5\) or \(2 * 3 * 5^2\)­
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Bunuel
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If the positive integer N has only three prime factors 2, 3, and 5 04 Jun 2024, 10:09
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envy5

gmatophobia

Bunuel
If the positive integer N has only three prime factors 2, 3, and 5, what is the value of N ?

(1) N has a total of 12 factors
(2) N > 100
Given

N has three prime factors

Statement 1

(1) N has a total of 12 factors

Possible values of N = \(2^2 * 3 * 5\) , \(2 * 3^2 * 5\) or \(2 * 3 * 5^2\)

As we can have multiple values of N, the statement is not sufficient and we can eliminate A and D.

Statement 2

(2)N > 100

Insufficient, eliminate B.

Combined

The statements combined we can eliminate the possibility of N = \(2^2 * 3 * 5\) and N = \(2 * 3^2 * 5\) as both of these numbers are less than 100.

Hence N = \(2 * 3 * 5^2\)

Sufficient.

Option C
­Hi would you be able to give an explanation/rule that gives the result of (1) N has a total of 12 factors - Possible values of N = \(2^2 * 3 * 5\) , \(2 * 3^2 * 5\) or \(2 * 3 * 5^2\)­
Show more
­
Finding the Number of Factors of an Integer

First, make the prime factorization of an integer \(n = a^p * b^q * c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\), and \(p\), \(q\), and \(r\) are their respective powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and \(n\) itself.

Example: Finding the number of all factors of 450: \(450 = 2^1 * 3^2 * 5^2\)

The total number of factors of 450, including 1 and 450 itself, is \((1+1)(2+1)(2+1) = 2*3*3 = 18\) factors.

According to the above, given that N has only three prime factors 2, 3, and 5, and that N has a total of 12 factors, we'd have \(N = 2^p * 3^q * 5^r\) and \((p + 1)(q + 1)(r + 1) = 12\). Since 12 can be expressed as the product of three positive integers, each greater than 1, in only one way: 12 = 2*2*3, then one from p, q, and r is 2 and the other two are 1.­
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