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# If r is an integer, is s an integer?

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Manager
Joined: 23 Sep 2016
Posts: 94
If r is an integer, is s an integer?  [#permalink]

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22 Nov 2016, 08:51
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Difficulty:

45% (medium)

Question Stats:

66% (01:56) correct 34% (02:17) wrong based on 79 sessions

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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.
Math Expert
Joined: 02 Aug 2009
Posts: 7199
Re: If r is an integer, is s an integer?  [#permalink]

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22 Nov 2016, 16:42
1
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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Joined: 14 Oct 2015
Posts: 249
GPA: 3.57
If r is an integer, is s an integer?  [#permalink]

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28 Dec 2017, 09:37
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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28 Dec 2017, 09:52
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
Re: If r is an integer, is s an integer? &nbs [#permalink] 28 Dec 2017, 09:52
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