If m has the Smallest Prime Numbers as its Only Prime Factor GMAT Data Sufficiency

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Question: If m has the smallest prime numbers as its only prime factor, is \(3\sqrt m\) an integer?

\(m^2\) is divisible by 32
\(\sqrt{m}\) is divisible by 4

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

Given: m has the smallest prime number as its only prime factor
To find: Is \(3\sqrt m\) is an integer? (Yes/No)

S1: \(m^2\) is divisible by 32 which means m can be 16, 32, 64, 128 etc.
If m = 16 = \(2^4\), the smallest prime factor is 2 and only prime factor but \(3\sqrt {16}\) is not an integer
No
If m = 64 =\(2^6\), the smallest prime number is 2 and only prime factor and \(3\sqrt {64}\) is 4 and it is an integer
Yes
Inconsistent answers
Not sufficient

S2: \(\sqrt m\) is divisible by 4 which means m = 16, 64, etc
Use example used in S1
Inconsistent answers
Not sufficient
S(1) + S(2) gives m = 32, 64 etc and by solving these values of n, you get different answers
Not sufficient

Correct Option: E

Approach Solution (2):

Given:

Positive integer m
The only prime factor of m = the smallest prime number
- The smallest prime number is 2
So, m = 2n , where n is a positive integer

Analyze Statement 1 independent

m2 is divisible by 32
m2 = 22n
32 = 25
Since m2 is divisible by 32, m2 must be greater than or equal to 32

\(\Rightarrow 2^{2n} \geq 2^5\)
\(\Rightarrow 2n \geq 5\)
\(\Rightarrow n \geq 2.5\)

Since n is an integer, the minimum possible value of n is 3.
So possible values of n = {3, 4, 5, 6 . . . }
Thus n may or may not be a multiple of 3

Step 4 : Analyze Statement 2 independent

\(\sqrt m \) is divisible by 4
\(\Rightarrow \sqrt m = 2 \frac{n}{2}\)
\(\Rightarrow 4=2^2\)

Since √m is divisible by 4, √m must be greater than or equal to 4
\(\Rightarrow 2 \frac{n}{2} \geq 2^2\)
\(\Rightarrow \frac {n}{2} \geq 2\)
\(\Rightarrow n \geq 4\)

So possible values of n = {4, 5, 6 . . . }
Thus n may or may not be a multiple of 3

Statement 2 alone is not sufficient to answer the question.

Step 5: Analyze Both Statements Together (if needed)

From Statement 1: n = {3, 4, 5, 6 . . . }
From Statement 2: n = {4, 5, 6 . . . }
By combining both statements: n = {4, 5, 6 . . . }

We’ve still not been able to determine if n is a multiple of 3.

So, even the two statements together are not sufficient to answer the question.

Correct Option: E

Approach Solution (3):

S1: According to statement 1, m can be 16, 32, 64, 128 etc.
If m is 16 = 2^4 = the smallest prime factor is 2 but cube root of 16 is not an integer
but in case of 64, the smallest prime factor is 2 and the cube root of 64 is an integer
So, we are getting inconsistent answers.
Hence insufficient

S2: According to statement 2, m can be 16, 64, etc.
It is same as statement 1
Hence insufficient

Correct Option: E

“If m has the smallest prime numbers as its only prime factor, is \(3\sqrt m\) an integer?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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If m has the Smallest Prime Numbers as its Only Prime Factor GMAT Data Sufficiency

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