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If n is a positive integer, is 361 a factor of n? 1) 76 is the grea

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If n is a positive integer, is 361 a factor of n? 1) 76 is the grea [#permalink]

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New post 13 Mar 2017, 23:50
2
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A
B
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D
E

Difficulty:

  65% (hard)

Question Stats:

58% (02:16) correct 42% (02:55) wrong based on 53 sessions

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If n is a positive integer, is 361 a factor of n?

1) 76 is the greatest common divisor of 380 and n
2) 5,776 is the least common multiple of 304 and n

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Re: If n is a positive integer, is 361 a factor of n? 1) 76 is the grea [#permalink]

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New post 14 Mar 2017, 02:42
1
1
If n is a positive integer, is 361 a factor of n?

1) 76 is the greatest common divisor of 380 and n
2) 5,776 is the least common multiple of 304 and n

statement 1
factors of 380 =2*2*5*19
factors of 76=2*2*19

since GCD is 76
so n could be 76 as well as
Statement 1 not sufficient

Statement 2
5776=2^4 * 19^2
304=2^4*19
since 5776 is the LCM of n and 304
so n must be of the form
2^4*19^2*p^a*q^b ........
and 361 is nothing but 19*19
so statement 2 is sufficient
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Re: If n is a positive integer, is 361 a factor of n? 1) 76 is the grea [#permalink]

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New post 15 Mar 2017, 05:44
St 1: let n =76 and 1444. the statement will hold true but 361 will not be the factor for 76 but will be the factor for 1444. INSUFFICIENT

St 2: 304 = 19*16
5776 = 19*19*16
therefore n should have a factor 19*19 to make the LCM 5776. ANSWER

option B
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Re: If n is a positive integer, is 361 a factor of n? 1) 76 is the grea [#permalink]

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New post 27 Mar 2017, 18:25
ziyuen wrote:
If n is a positive integer, is 361 a factor of n?

1) 76 is the greatest common divisor of 380 and n
2) 5,776 is the least common multiple of 304 and n


OFFICIAL SOLUTION



The greatest common divisor of the integers is the least value of the exponents, and the least common multiple of the integers is the maximum value of the quotients.

Since there is 1 variable (n) in the original condition, D is most likely to be the answer.

In the case of con 1), from \(361=19^2\), \(76=2^2*19\), and \(380=2^2*5*19\), the greatest common divisor of 380 and n is 76.Since the greatest common divisor of the integers is the least value of the exponents, \(n=2^2*19\), \(2^2*3*19\),…., hence it is not unique, and not sufficient.

In the case of con 2), from \(5,776=2^4*19^2\), \(304=2^4*19\), the least common multiple of n and \(2^4*19\) is \(2^4*19^2\). Since the least common multiple of the integers is the maximum value of the exponents, n should always have \(19^2\), hence always yes, and it is sufficient. Therefore, the answer is B.
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Re: If n is a positive integer, is 361 a factor of n? 1) 76 is the grea [#permalink]

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New post 27 Mar 2017, 18:58
I was completely stumped by this question. Any idea how to factorize 361, 76, and 5776 in a better approach? Please help.
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Re: If n is a positive integer, is 361 a factor of n? 1) 76 is the grea [#permalink]

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New post 27 Mar 2017, 21:05
ziyuen wrote:
If n is a positive integer, is 361 a factor of n?

1) 76 is the greatest common divisor of 380 and n
2) 5,776 is the least common multiple of 304 and n


Key here is to identify 361 - \(19^2\). It took 1 min for me to figure out this...
Re: If n is a positive integer, is 361 a factor of n? 1) 76 is the grea   [#permalink] 27 Mar 2017, 21:05
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