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If r is an integer, is s an integer?

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If r is an integer, is s an integer?  [#permalink]

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New post 22 Nov 2016, 09:51
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If r is an integer, is s an integer?

(1) The average (arithmetic mean) of r, s, and 2s + 3 is not an integer.
(2) The average (arithmetic mean) of r, s, and s + 1 is r.
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Re: If r is an integer, is s an integer?  [#permalink]

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New post 22 Nov 2016, 17:42
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SW4 wrote:
If r is an integer, is s an integer?

(1) The average (arithmetic mean) of r, s, and 2s + 3 is not an integer.
(2) The average (arithmetic mean) of r, s, and s + 1 is r.


Let's check the statements
1) \(\frac{r+s+2s+3}{3}=\frac{r+3s+3}{3}\) is not an integer..
This just tells us
A) r/3 is not an integer or r is not a multiple of 3, then s can be an integer OR
B) s itself is not an integer
Insuff

2) \(\frac{r+s+s+1}{3}=r......r+2s+1=3r.....2s+1=2r......2r-2s=1\)
Now r is integer so 2r is EVEN....
If s is an integer, 2s will also be EVEN and difference between two EVENs cannot be 1, it has to be a multiple of 2
So s is not an integer..
Suff

B
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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


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If r is an integer, is s an integer?  [#permalink]

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New post 28 Dec 2017, 10:37
SW4 wrote:
If r is an integer, is s an integer?

(1) The average (arithmetic mean) of r, s, and 2s + 3 is not an integer.
(2) The average (arithmetic mean) of r, s, and s + 1 is r.


It should be B
1- Let us calculate Arithmetic mean.

\(\frac{r+s+2s}{3}\)

Simplifying further.

\(\frac{r+3s}{3}\)

or \(\frac{r}{3}+s\) is not an integer. This does not give us much about \(s\) being an integer. \(\frac{r}{3}\) could be an integer if \(r\) is a multiple of 3, or may not be an integer which makes it impossible to accurately determine whether \(s\) is an integer.

2- Creating an expression for the Arithmetic mean.

\(\frac{r+s+s+1}{3} = r\)

\(r+2s+1 = 3r\)

\(2s = 2r - 1\)

\(s = 2r - \frac{1}{2}\)

We know \(r\) (and consequently \(2r\)) is an integer so \(s\) being \(\frac{1}{2}\) less than \(2r\) is not an integer.
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Re: If r is an integer, is s an integer?  [#permalink]

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New post 28 Dec 2017, 10:52
SW4 wrote:
If r is an integer, is s an integer?

(1) The average (arithmetic mean) of r, s, and 2s + 3 is not an integer.
(2) The average (arithmetic mean) of r, s, and s + 1 is r.


Statement 1: implies \(r+s+2s+3=3*N.I\) (non integer)

\(=> r= 3*N.I-3s-3 => 3*N.I-Integer\) (\(s\) is a integer so \(3s\) will be integer hence \(3s-3\) will also be an integer)

if \(N.I=\frac{1}{3}\) ,\(\frac{2}{3}\) etc then \(3*N.I=integer\), hence \(r=Integer-Integer= Integer\)

but if \(N.I=\frac{1}{2}, \frac{3}{5}\) etc. then \(3*N.I= N.I\), hence \(r=N.I-Integer = N.I\). Insufficient

Statement 2: implies \(r+s+s+1=3r => 2r=2s+1\)

\(=> r=s+\frac{1}{2}=Integer+N.I\) (sum of integer and non integer cannot be an integer)

Hence \(r=\)non integer. Sufficient

Option B
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Re: If r is an integer, is s an integer? &nbs [#permalink] 28 Dec 2017, 10:52
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