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# What is the remainder when the positive integer x is divided by 9?

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What is the remainder when the positive integer x is divided by 9?  [#permalink]

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29 Oct 2018, 05:43
2
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Difficulty:

55% (hard)

Question Stats:

64% (01:24) correct 36% (01:45) wrong based on 90 sessions

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What is the remainder when the positive integer x is divided by 9?
(1) x + 34 is a multiple of 18
(2) x is a multiple of 11
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Re: What is the remainder when the positive integer x is divided by 9?  [#permalink]

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29 Oct 2018, 06:05
rencsee wrote:
What is the remainder when the positive integer x is divided by 9?
(1) x + 34 is a multiple of 18
(2) x is a multiple of 11

Question: What is the remainder when the positive integer x is divided by 9?

Statement 1: x + 34 is a multiple of 18

i.e. x+34 may be 36, 54, 72...

i.e. x may be 2, 20, 38 ...

i.e remainder on dividing x by 9 = 2 Always

SUFFICIENT

Statement 2: x is a multiple of 11

i.e. x may be 11, 22, 33 ...

i.e remainder on dividing x by 9 = 2, 4, 6 etc

NOT SUFFICIENT

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Re: What is the remainder when the positive integer x is divided by 9?  [#permalink]

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29 Oct 2018, 09:57
1
What is the remainder when the positive integer x is divided by 9?
(1) x + 34 is a multiple of 18
(2) x is a multiple of 11

Stmnt1:
checking for all values of x where it is divisible by 18 , we get x= 20,38,54 , where x when divided by 9 gives a same remainder i.e. 2 hence sufficient

Stmnt 2:
x is a multiple of 11

x=11,22,33,44, when x divided by 9 gives remainder (2,4,6,8) so insufficient

Hence A is the correct answer
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Re: What is the remainder when the positive integer x is divided by 9?  [#permalink]

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30 Oct 2018, 14:35
1
rencsee wrote:
What is the remainder when the positive integer x is divided by 9?
(1) x + 34 is a multiple of 18
(2) x is a multiple of 11

$$\left\{ \matrix{ \,\,1\, \le \,x = 9Q + r\,\,,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr \,\,0 \le r \le 8\,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\, - 2\,\, \le r - 2\,\, \le \,\,6\,\,\,\left( * \right) \hfill \cr} \right.$$

$$? = r$$

$$\left( 1 \right)\,\,\,\frac{{x + 34}}{{18}} = \operatorname{int} \,\,\,\, \Rightarrow \,\,\,\,\operatorname{int} = \,\frac{{9Q + r + 36 - 2}}{{18}} = \frac{{9Q + \left( {r - 2} \right)}}{{18}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\{ \begin{gathered} \,Q\,\,{\text{even}} \hfill \\ \,r - 2 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2 \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,\,\frac{x}{{11}} = \operatorname{int} \,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,x = 11\,\,\,\,\left[ {Q = 1,\,\,r = 2} \right]\,\,\,\,\, \Rightarrow \,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{2}}\,\, \hfill \\ \,{\text{Take}}\,\,x = 22\,\,\,\,\left[ {Q = 2,\,\,r = 4} \right]\,\,\,\,\, \Rightarrow \,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{4}}\,\, \hfill \\ \end{gathered} \right.\,\,$$

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

Regards,
Fabio.
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Re: What is the remainder when the positive integer x is divided by 9?  [#permalink]

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01 Nov 2018, 23:30
What is the remainder when the positive integer x is divided by 9?
(1) x + 34 is a multiple of 18
(2) x is a multiple of 11

1. In other words, x-2 is a multiple of 18. Hence, x leaves remainder 2 with 18. Hence, x leaves remainder 2 with 9 also. Sufficient.

2. Too many cases possible. Doesn't help. Insufficient.

A.
Re: What is the remainder when the positive integer x is divided by 9?   [#permalink] 01 Nov 2018, 23:30
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