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

2

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.

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

2

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.

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.

Re: n is divisible by 14 and 3. Which of the following statement
[#permalink]
26 May 2022, 15:53

1

We have to examine each choice to confirm must-validity:

A. 12 is a factor of n.

is not true because n is divisble by 14

B. 21 is a factor of n.

True because the least common multiple between 14 and 3 is 42, therefore 21 is a factor of 42

C. n is a multiple of 42.

LCF is 42, true

Answer is (B & C)

A. 12 is a factor of n.

is not true because n is divisble by 14

B. 21 is a factor of n.

True because the least common multiple between 14 and 3 is 42, therefore 21 is a factor of 42

C. n is a multiple of 42.

LCF is 42, true

Answer is (B & C)

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Re: n is divisible by 14 and 3. Which of the following statement
[#permalink]
14 Jun 2024, 03:40

1

↧↧↧ Detailed Video Solution to the Problem Series ↧↧↧

n is divisible by 14 and 3

=> n = LCM(3,14)*k (where k is an integer) = 42k [Watch this video to MASTER LCM and GCD]

I. 12 is a factor of n.

n = 42k = 6*7k

Now, 12 will be a factor of n ONLY when k is even, which might not be true always

=> FALSE for MUST be true

II. 21 is a factor of n.

n = 42k = 21*2k

=> 21 is a factor of n

=> TRUE

III. n is a multiple of 42.

n = 42k

=> n is a multiple of 42

=> TRUE

So, Answer will be B, C

Hope it helps!

Watch the following video to learn the basics of Factors and Multiples

n is divisible by 14 and 3

=> n = LCM(3,14)*k (where k is an integer) = 42k [Watch this video to MASTER LCM and GCD]

I. 12 is a factor of n.

n = 42k = 6*7k

Now, 12 will be a factor of n ONLY when k is even, which might not be true always

=> FALSE for MUST be true

II. 21 is a factor of n.

n = 42k = 21*2k

=> 21 is a factor of n

=> TRUE

III. n is a multiple of 42.

n = 42k

=> n is a multiple of 42

=> TRUE

So, Answer will be B, C

Hope it helps!

Watch the following video to learn the basics of Factors and Multiples

gmatclubot

Re: n is divisible by 14 and 3. Which of the following statement [#permalink]

14 Jun 2024, 03:40
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