How many factors does the number X have?

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How many factors does the number X have?

1) X is divisible by 47
2) X lies between 100 and 150, inclusive.

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


OA C

Source: e-GMAT

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by Jay@ManhattanReview » Sun Oct 14, 2018 9:02 pm

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BTGmoderatorDC wrote:How many factors does the number X have?

1) X is divisible by 47
2) X lies between 100 and 150, inclusive.

OA C

Source: e-GMAT
Question: How many factors does the number X have?

Let's take each statement one by one.

1) X is divisible by 47.

Certainly insufficient. If X = 47, then the number of factors of X (1 and 47)= 2; however, if say X = 94, then the number of factors of X (1, 2, 47, and 94) = 4. No unique answer. Insufficient.

2) X lies between 100 and 150, inclusive.

Certainly insufficient. X can be any number between 100 and 150, inclusive. So there are many possible numbers of factors.

(1) and (2) together

The only possible value of X = 3*47 = 141. The number of factors of X (1, 3, 47, and 141) = 4. Unique answer. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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by Brent@GMATPrepNow » Mon Oct 15, 2018 5:58 am

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BTGmoderatorDC wrote:How many factors does the number X have?

1) X is divisible by 47
2) X lies between 100 and 150, inclusive.
Target question: How many factors does the number X have?

Statement 1: X is divisible by 47
Let's TEST some values.
There are several values of X that satisfy statement 1. Here are two:
Case a: X = 47. The factors of 47 are {1, 47}. So, the answer to the target question is X has 2 factors
Case b: X = 94. The factors of 94 are {1, 2, 47, 94}. So, the answer to the target question is X has 4 factors
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: X lies between 100 and 150, inclusive.
There are several values of X that satisfy statement 2. Here are two:
Case a: X = 47. The factors of 101 are {1, 101}. So, the answer to the target question is X has 2 factors
Case b: X = 106. The factors of 106 are {1, 2, 53, 106}. So, the answer to the target question is X has 4 factors
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that X is divisible by 47. So, some possible values of X are: 47, 94, 141, 188, 235, etc
Statement 2 tells us that X lies between 100 and 150, inclusive.
Only 141 satisfies BOTH statements, so it must be the case that X = 141.
Now that we know the value of X, we COULD determine how many factors it has (but we'd never waste time actually doing so on test day)
Since we COULD answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
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BTGmoderatorDC wrote:How many factors does the number X have?

1) X is divisible by 47
2) X lies between 100 and 150, inclusive.

Source: e-GMAT
\[?\,\,:\,\,\,\,\# \,\,{\text{of}}\,\,{\text{factors}}\,\,{\text{of}}\,\,{\text{X}}\]
\[\left( 1 \right)\,\,\,\,\,\frac{X}{{47}} = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,X = 0\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{infinite}}\,\,\,\,\,\,\left[ {\,{\text{every}}\,\,{\text{nonzero}}\,\,{\text{integer}}\,} \right]\, \hfill \\
\,{\text{Take}}\,\,X = 47\,\,\,\,\left( {{\text{prime}}} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,4\,\,\,\,\,\,\,\,\left[ {\, - 47\,,\, - 1\,,\,\,1\,,\,\,47\,} \right] \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,100 \leqslant X \leqslant 150\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,X = 101\,\,\,\left( {{\text{prime}}} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,4\,\,\,\,\,\,\,\,\left[ {\, - 101\,,\, - 1\,,\,\,1\,\,,\,\,101\,} \right] \hfill \\
\,{\text{Take}}\,\,X = 100\,\,\,\, \Rightarrow \,\,\,?\,\, > \,\,4\, \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\,X = 47 \cdot 3\,\,\,\,\,\, \Rightarrow \,\,\,\,\,?\,\,\,{\text{unique}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\text{SUFF}}.\,\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: -47 is a factor of 47, because both are integers and 47/(-47) is an integer.
This is not only mathematically perfect, but also GMAT-definition-of-factor accepted.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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