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If k is a positive integer and the greatest common divisor of k and 45

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If k is a positive integer and the greatest common divisor of k and 45  [#permalink]

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New post 16 Nov 2016, 04:55
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Question Stats:

72% (01:37) correct 28% (01:49) wrong based on 123 sessions

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If k is a positive integer and the greatest common divisor of k and 45 is 15, then the greatest common divisor of k and 900 may be any of the following EXCEPT

A. 15
B. 45
C. 60
D. 150
E. 300

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If k is a positive integer and the greatest common divisor of k and 45  [#permalink]

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New post 16 Nov 2016, 06:26
Bunuel wrote:
If k is a positive integer and the greatest common divisor of k and 45 is 15, then the greatest common divisor of k and 900 may be any of the following EXCEPT

A. 15
B. 45
C. 60
D. 150
E. 300


IMO B


A. 15== GCD(15,45)= 15 GCD(15,900)= 15
B. 45== GCD(45,45)= 45 GCD(45,900)= 45 Violates the given GCD(k,45)=15
C. 60== GCD(60,45)= 15 GCD(60,900)= 60
D. 150== GCD(150,45)= 15 GCD(150,900)=150
E. 300== GCD(300,45)= 15 GCD(300,900)= 300
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Re: If k is a positive integer and the greatest common divisor of k and 45  [#permalink]

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New post 14 Mar 2018, 19:41
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Hi All,

This question is ultimately about "prime factorization" (the idea that a positive integer is either prime or can be "broken down" into the product of a bunch of primes).

We're told that the greatest common divisor of K and 45 is 15.

Since 45 = (3)(3)(5)
and 15 = (3)(5)
then K, at the minimum is (3)(5). It might have OTHER prime factors as well, BUT it can't have another 3. Here's why…

If K = (2)(3)(5) = 30, then the greatest common divisor of 30 and 45 is 15, so the above statement holds true.
If K = (3)(3)(5) = 45, then the greatest common divisor of 45 and 45 is 45, and the above statement doesn't make sense.

We're asked which of the following could be the greatest common divisor of K and 900 EXCEPT….

900 = (3)(3)(2)(2)(5)(5)
K = (3)(5)(possibly some other primes other than 3)

We can now use the answer choices to eliminate the possibilities:

A = 15 = (3)(5) IS possible
B = 45 = (3)(3)(5) is NOT possible
C = 60 = (3)(5)(2)(2) IS possible
D = 150 = (3)(5)(2)(5) IS possible
E = 300 = (3)(5)(2)(2)(5) IS possible

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Re: If k is a positive integer and the greatest common divisor of k and 45 &nbs [#permalink] 14 Mar 2018, 19:41
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