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# What is the remainder when the positive integer n is divided by 2?

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What is the remainder when the positive integer n is divided by 2?  [#permalink]

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29 Oct 2018, 06:46
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50% (01:29) correct 50% (02:01) wrong based on 84 sessions

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What is the remainder when the positive integer n is divided by 2?
(1) When n is divided by 13, the remainder is 3
(2) n + 2 is a multiple of 7
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Re: What is the remainder when the positive integer n is divided by 2?  [#permalink]

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29 Oct 2018, 07:27
rencsee wrote:
What is the remainder when the positive integer n is divided by 2?
(1) When n is divided by 13, the remainder is 3
(2) n + 2 is a multiple of 7

Question: What is the remainder when the positive integer n is divided by 2?

To answer this we only need to know whether n is Even or Odd

Question REPHRASED: Is n even or Odd?

Statement 1: When n is divided by 13, the remainder is 3
n may be 16 or 29 or 42 or 55 hence
NOT SUFFICIENT

Statement 2: n + 2 is a multiple of 7
n+2 may be 7 or 14 or 21 etc
n may be 5 or 12 or 19 etc
NOT SUFFICIENT

Combining the two statements
n = 13a+3 = 7b-2
i.e. b = (13a+5)/7 = 2a +(5-a)/7

for (5-a) to be divisible by 7, a should be 12 or 19 etc

i.e. n may be (13a+3) = 159 or 250 etc (i.e. Sometimes even and some other time ODD hence
NOT SUFFICIENT

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Re: What is the remainder when the positive integer n is divided by 2?  [#permalink]

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30 Oct 2018, 15:08
rencsee wrote:
What is the remainder when the positive integer n is divided by 2?
(1) When n is divided by 13, the remainder is 3
(2) n + 2 is a multiple of 7

$$n \geqslant 1\,\,\,\left( * \right)$$

$$n\,\,\mathop = \limits^? \,\,{\text{even}}\,\,\,\,\,\left[ \begin{gathered} n\,\,{\text{even}}\,\, \Rightarrow \,\,\,{\text{remainder}} = 0\, \hfill \\ \,n\,\,{\text{odd}}\,\, \Rightarrow \,\,\,{\text{remainder}} = 1 \hfill \\ \end{gathered} \right]$$

$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{ \,n = 13Q + 3\,\,\,,\,\,\,\left( * \right)\,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr \,n + 2 = 7M\,\,\,,\,\,\,\left( * \right)\,\,\,M \ge 1\,\,{\mathop{\rm int}} \, \hfill \cr} \right.$$

$$7M = n + 2 = 13Q + 5\,\,\,\,\,\, \Rightarrow \,\,\,13Q + 5\,\,{\text{multiple}}\,\,{\text{of}}\,\,7\,\,\,$$

$$\left\{ \matrix{ \,Q = 5 \,\, \Rightarrow \,\,\,13Q + 5 = 70 = 7 \cdot 10 = n + 2\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr \,Q = 12 \,\, \Rightarrow \,\,\,13Q + 5 = 161 = 7 \cdot 23 = n + 2\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$

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 n is divided by 2?  [#permalink]

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01 Nov 2018, 23:36
What is the remainder when the positive integer n is divided by 2?
(1) When n is divided by 13, the remainder is 3
(2) n + 2 is a multiple of 7

1. For n=16, remainder is 0. For n=29, remainder is 1. Insufficient.
2. For n=5, remainder is 1. For n=12, remainder is 0. Insufficient.
Together, still many cases will be formed, n could be even or odd. Insufficient.
E.
Re: What is the remainder when the positive integer n is divided by 2?   [#permalink] 01 Nov 2018, 23:36
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