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Retired Moderator

Joined: **07 Jun 2014 **

Posts: **4805**

WE:**Business Development (Energy and Utilities)**

n is divisible by 14 and 3. Which of the following statement
[#permalink]
12 Aug 2018, 16:05

Expert Reply

Question Stats:

n is divisible by 14 and 3. Which of the following statements must be true?

Indicate all such statements.

A. 12 is a factor of n.

B. 21 is a factor of n.

C. n is a multiple of 42.

_________________

Indicate all such statements.

A. 12 is a factor of n.

B. 21 is a factor of n.

C. n is a multiple of 42.

_________________

Sandy

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Retired Moderator

Joined: **07 Jun 2014 **

Posts: **4805**

WE:**Business Development (Energy and Utilities)**

Re: n is divisible by 14 and 3. Which of the following statement
[#permalink]
15 Aug 2018, 05:19

Expert Reply

Explanation

Since n is divisible by 14 and 3, n contains the prime factors of both 14 and 3, which are 2, 7, and 3. Thus, any numbers that can be constructed using only these prime factors (no additional factors) are factors of n.

Since \(12 = 2 \times 2 \times 3\), you cannot make 12 by multiplying the prime factors of n (you would need one more 2). However, you can construct 21 by multiplying two of the known prime factors of n (\(7 \times 3 = 21\)), so the second statement is true.

Finally, n must be at least 42 (\(= 2 \times 7 \times 3\), the least common multiple of 14 and 3), so n is definitely a multiple of 42. That is, n can only be 42, 84, 126, etc.

_________________

Sandy

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Since n is divisible by 14 and 3, n contains the prime factors of both 14 and 3, which are 2, 7, and 3. Thus, any numbers that can be constructed using only these prime factors (no additional factors) are factors of n.

Since \(12 = 2 \times 2 \times 3\), you cannot make 12 by multiplying the prime factors of n (you would need one more 2). However, you can construct 21 by multiplying two of the known prime factors of n (\(7 \times 3 = 21\)), so the second statement is true.

Finally, n must be at least 42 (\(= 2 \times 7 \times 3\), the least common multiple of 14 and 3), so n is definitely a multiple of 42. That is, n can only be 42, 84, 126, etc.

_________________

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gmatclubot

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