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Is x/11 an integer?

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Is x/11 an integer?  [#permalink]

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New post 09 Oct 2018, 13:36
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Is x/11 an integer?

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

Source: http://www.GMATH.net

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

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Is x/11 an integer?  [#permalink]

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New post 09 Oct 2018, 15:01
1
fskilnik wrote:
Is x/11 an integer?

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

Source: http://www.GMATH.net


5x/11 = 1
x = 11/5

11/5 * 1/11 = 1/5 not integer.

If x = 11 then yes

Statement 1 is insufficient.

2) follow the same logic.

We get two different answers insufficient

Combine (1) and (2)

x has to be a multiple of 11

Sufficient.

C

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Re: Is x/11 an integer?  [#permalink]

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New post 09 Oct 2018, 17:57
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Is x/11 an integer?

(1) 5x/11 is an integer
Let \(\frac{5x}{11}=y.......5x=11y.......x=\frac{11y}{5}\)
If y is a multiple of 5...yes
Otherwise No
Insufficient

(2) 7x/11 is an integer
Let \(\frac{7x}{11}=z......7x=11z.......x=\frac{11z}{7}\)
If z is a multiple of 7..yes
Otherwise No
Insufficient


Combined
\(\frac{11y}{5}\)=\(\frac{11z}{7}\).....7y=5z
So y has to be a multiple of 5 and z a multiple of 7..
Thus x is an integer
Sufficient

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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Is x/11 an integer?  [#permalink]

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New post 10 Oct 2018, 05:14
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fskilnik wrote:
Is x/11 an integer?

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

\(\frac{x}{{11}}\,\,\mathop = \limits^? \,\,\operatorname{int}\)

\(\left( 1 \right)\,\,\,\frac{{5x}}{{11}}\,\, = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = \frac{{11}}{5}\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,\frac{1}{5}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)

\(\left( 2 \right)\,\,\,\frac{{7x}}{{11}}\,\, = \operatorname{int} \,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,x = 0\,\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,x = \frac{{11}}{7}\,\,\,\, \Rightarrow \,\,\,\,\frac{x}{{11}} = \,\,\frac{1}{7}\,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\)


\(\left( {1 + 2} \right)\,\,\,\,\frac{x}{{11}}\,\,\, = \,\,\,\frac{{15x}}{{11}} - \frac{{14x}}{{11}}\,\,\, = \,\,\,3 \cdot \left( {\frac{{5x}}{{11}}} \right) - 2 \cdot \left( {\frac{{7x}}{{11}}} \right)\,\,\, = \,\,\,3 \cdot \operatorname{int} - 2 \cdot \operatorname{int} \,\,\, = \,\,\,\operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle\)


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

Regards,
Fabio.
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Re: Is x/11 an integer?  [#permalink]

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New post 10 Oct 2018, 09:50
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fskilnik wrote:
Is x/11 an integer?

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

Source: http://www.GMATH.net



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

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Re: Is x/11 an integer?   [#permalink] 10 Oct 2018, 09:50
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