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Is x/11 an integer?

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Is x/11 an integer?

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Is x/11 an integer?

(1) 5x/11 is an integer
(2) 7x/11 is an integer

Answer: __(C)____
Difficulty Level: 650 - 700
Source: http://www.GMATH.net

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Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or https://GMATH.com.br (Portuguese version)
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fskilnik@GMATH wrote:
Is x/11 an integer?

(1) 5x/11 is an integer
(2) 7x/11 is an integer

Source: www.GMATH.net
\[\frac{x}{{11}}\,\,\mathop = \limits^? \,\,\operatorname{int} \]
\[\left( 1 \right)\,\,\,\frac{{5x}}{{11}}\,\, = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = \frac{{11}}{5}\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,\frac{1}{5}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,\frac{{7x}}{{11}}\,\, = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = \frac{{11}}{7}\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,\frac{1}{7}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\,\frac{x}{{11}}\,\,\, = \,\,\,\frac{{15x}}{{11}} - \frac{{14x}}{{11}}\,\,\, = \,\,\,3 \cdot \left( {\frac{{5x}}{{11}}} \right) - 2 \cdot \left( {\frac{{7x}}{{11}}} \right)\,\,\, = \,\,\,3 \cdot \operatorname{int} \, - \,\,2 \cdot \operatorname{int} \,\,\, = \,\,\,\operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]

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

Regards,
Fabio.

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Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or https://GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

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fskilnik@GMATH wrote:
Is x/11 an integer?

(1) 5x/11 is an integer
(2) 7x/11 is an integer
Target question: Is x/11 an integer?

Statement 1: 5x/11 is an integer
Let's TEST some values
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 11. Notice that 5x/11 = 5(11)/11 = 5, which is an integer. In this case, x/11 = 11/11 = 1. So, the answer to the target question is YES, x/11 IS an integer
Case b: x = 4.4. Notice that 5x/11 = 5(4.4)/11 = 22/11 = 2, which is an integer. In this case, x/11 = 4.4/11 = 0.4. So, the answer to the target question is NO, x/11 in NOT an integer
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 7x/11 is an integer
Let's TEST some values
There are several values of x that satisfy statement 2. Here are two:
Case a: x = 11. Notice that 7x/11 = 7(11)/11 = 7, which is an integer. In this case, x/11 = 11/11 = 1. So, the answer to the target question is YES, x/11 IS an integer
Case b: x = 22/7 Notice that 7x/11 = 7(22/7)/11 = 22/11 = 2, which is an integer. In this case, x/11 = (22/7)/11 = 2/7. So, the answer to the target question is NO, x/11 in NOT an integer
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
We can use a nice rule that says: If j is divisible by n, and k is divisible by n, then (j - k) is divisible by n

Statement 1 tells us that 5x is divisible by 11
If 5x is divisible by 11, then (3)(5x) is divisible by 11
In other words, 15x is divisible by 11

Statement 2 tells us that 7x is divisible by 11
If 7x is divisible by 11, then (2)(7x) is divisible by 11
In other words, 14x is divisible by 11

If 15x is divisible by 11 and 14x is divisible by 11, we can apply the above rule to conclude that (15x - 14x) is divisible by 11
In other words, x is divisible by 11
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

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