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# What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer

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Joined: 04 Jan 2015
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What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer  [#permalink]

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07 Jun 2018, 21:44
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Difficulty:

85% (hard)

Question Stats:

47% (02:12) correct 53% (02:04) wrong based on 87 sessions

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What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer and a ≥ 3?

1. ‘a’ is a prime number greater than 2.
2. Both GCD (3, a) and LCM (3, a) are factors of the number 30.

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Re: What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer  [#permalink]

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08 Jun 2018, 07:25
EgmatQuantExpert wrote:
What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer and a ≥ 3?

1. ‘a’ is a prime number greater than 2.
2. Both GCD (3, a) and LCM (3, a) are factors of the number 30.

Option-1 - "a" can have any number greater than or equal to 3. So we can't determine the exact GCD(3,a,12). Insufficient.
Option-2 - Here "a" can have values such as 5, 6 and 10. In all cases GCD(3,a) and LCM(3,a) are factors of 30. So we can't derive exact value of GCD(3,a,12). Insufficient.

Now, combining both options we will have only one value of "a=5" as 5 is the only prime number greater than 2. So we can find out the exact value of GCD now.
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What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer  [#permalink]

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08 Jun 2018, 21:17
jackspire wrote:
EgmatQuantExpert wrote:
What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer and a ≥ 3?

1. ‘a’ is a prime number greater than 2.
2. Both GCD (3, a) and LCM (3, a) are factors of the number 30.

Option-1 - "a" can have any number greater than or equal to 3. So we can't determine the exact GCD(3,a,12). Insufficient.
Option-2 - Here "a" can have values such as 5, 6 and 10. In all cases GCD(3,a) and LCM(3,a) are factors of 30. So we can't derive exact value of GCD(3,a,12). Insufficient.

Now, combining both options we will have only one value of "a=5" as 5 is the only prime number greater than 2. So we can find out the exact value of GCD now.

What if we consider a=3? then GCD of (3,3) =3 and LCm of (3,3) =3 which is a factor of 30. Hence answer should be E in my opinion.
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Re: What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer  [#permalink]

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11 Jun 2018, 05:22

Solution

Given:
• a is an integer
• a ≥ 3

To find:
• The GCD of (3, a, 12)

Analysing Statement 1
• As per the information given in statement 1, ‘a’ is a prime number greater than 2
o Therefore, ‘a’ can have values like 3, 5, 7, 11 etc

• If a = 3, GCD (3, a, 12) = GCD (3, 3, 12) = 3
• If a = 5, GCD (3, a, 12) = GCD (3, 5, 12) = 1
• If a = 7, GCD (3, a, 12) = GCD (3, 7, 12) = 1

We can see, for different values of a, we are getting different GCD values

Hence, statement 1 is not sufficient to answer

Analysing Statement 2
• As per the information given in statement 2, both GCD (3, a) and LCM (3, a) are factors of the number 30
o Factors of 30 = 1, 2, 3, 5, 6, 10, 15, and 30

• If a = 3, GCD (3, a) = GCD (3, 3) = 1 and LCM (3, 3) = 3
o Both 1 and 3 are factors of 30
o Also, GCD (3, a, 12) = GCD (3, 3, 12) = 3

• If a = 5, GCD (3, a) = GCD (3, 5) = 1 and LCM (3, 5) = 15
o Both 1 and 15 are factors of 30
o Also, (3, a, 12) = GCD (3, 5, 12) = 1

We can see a can have multiple possible values which satisfy the given statement, and the GCD of (3, a, 12) is different for different values of a

Hence, statement 2 is not sufficient to answer

Combining Both Statements
Even after combining the statements, we can say
• a can be 3 or 5, for which GCD is different – hence unique value of a cannot be determined

Hence, the correct answer is option E.

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Re: What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer  [#permalink]

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23 Jun 2018, 11:40
What is the GCD of 3,a,12, where a ≥ 3? This is the same as what is the GCD of 3 and a?
If a is a multiple of 3, answer will be 3. If a is not a multiple of 3, answer will be 1.

(1)
a is a prime greater than 2.
If a=3, GCD=3
If a=5, GCD=1
Not suff

(2)
a=3 and a=5 also fit here.

Re: What is the GCD of the numbers 3, a, and 12, if ‘a’ is an integer &nbs [#permalink] 23 Jun 2018, 11:40
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