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# If m and n are consecutive positive integers, is m greater than n ?

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If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 02:04
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If m and n are consecutive positive integers, is m greater than n ?

(1) m-1 and n+1 are consecutive positive integers.
(2) m is an even integer.

Data Sufficiency
Question: 86
Category: Arithmetic Properties of numbers
Page: 158
Difficulty: 650

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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 02:04
SOLUTION

If m and n are consecutive positive integers, is m greater than n?

(1) m-1 and n+1 are consecutive positive integers --> m>n, if m were less than n than m-1 (integer less than m) and n+1 (integer more than n) wouldn't be consecutive. Sufficient.

Or look at this in another way: stem says that the distance between m and n is 1. Now, if m<n then the distance between m-1 and n+1 would be 3 and they couldn't be consecutive as (1) states. Thus it must be true that m>n.

(2) m is an even integer. Clearly insufficient.

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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 11:36
6
The question states that $$m$$ &$$n$$ are consecutive positive integers. The question can then be boiled down to whether the order is $$m,n$$ or $$n,m$$.

(1) m-1 and n+1 are consecutive positive integers.

We know that $$m-1$$ and $$n+1$$are consecutive positive integers. If the order is $$m,n$$ then this statement clearly fails. If the order is $$n,m$$ then this statement is true, hence $$A$$ is sufficient.

(2) m is an even integer.

If $$m$$ is an even integer, $$n$$ must be odd. $$n$$ can still fall to the left or right of $$m$$, and therefore statement (2) is insufficient.

##### General Discussion
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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 05:12
1
St1: m-1 and n+1 are consecutive positive integers.

Case 1:
m is even, n is odd and m < n
Let m = 2, n = 3
m-1 = 1, n+1 = 4 -> not consecutive positive integers.

Case 2:
m is even, n is odd and m > n
Let m = 2, n = 1
m-1 = 1, n+1 = 2 -> consecutive positive integers.

Case 3:
m is odd, n is even and m < n
Let m = 3, n = 4
m-1 = 2, n+1 = 5 -> not consecutive positive integers.

Case 4:
m is odd, n is even and m > n
Let m = 3, n = 2
m-1 = 2, n+1 = 3 -> consecutive positive integers.

Not sufficient. Dow to B, C or E.

St2: m is an even integer. Not sufficient. If m = 2 and n = 1 -> m > n but if m = 2 and n = 3 -> m < n.

St 1 + St2: Not sufficient as illustrated by case 1 and case 2.

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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 08:56
2
Donnie84 wrote:
St1: m-1 and n+1 are consecutive positive integers.

Case 1:
m is even, n is odd and m < n
Let m = 2, n = 3
m-1 = 1, n+1 = 4 -> not consecutive positive integers.

Case 2:
m is even, n is odd and m > n
Let m = 2, n = 1
m-1 = 1, n+1 = 2 -> consecutive positive integers.

Case 3:
m is odd, n is even and m < n
Let m = 3, n = 4
m-1 = 2, n+1 = 5 -> not consecutive positive integers.

Case 4:
m is odd, n is even and m > n
Let m = 3, n = 2
m-1 = 2, n+1 = 3 -> consecutive positive integers.

Not sufficient. Dow to B, C or E.

St2: m is an even integer. Not sufficient. If m = 2 and n = 1 -> m > n but if m = 2 and n = 3 -> m < n.

St 1 + St2: Not sufficient as illustrated by case 1 and case 2.

I think you are making a mistake somewhere !! Well from the different cases you have illustrated it is clear that "A" is sufficient !! Because both cases where m-1,n+1 were consecutive you had m>n therefore statement "1" is clearly sufficient. Hope that helped
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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 10:09
Zatmah wrote:
Donnie84 wrote:
St1: m-1 and n+1 are consecutive positive integers.

Case 1:
m is even, n is odd and m < n
Let m = 2, n = 3
m-1 = 1, n+1 = 4 -> not consecutive positive integers.

Case 2:
m is even, n is odd and m > n
Let m = 2, n = 1
m-1 = 1, n+1 = 2 -> consecutive positive integers.

Case 3:
m is odd, n is even and m < n
Let m = 3, n = 4
m-1 = 2, n+1 = 5 -> not consecutive positive integers.

Case 4:
m is odd, n is even and m > n
Let m = 3, n = 2
m-1 = 2, n+1 = 3 -> consecutive positive integers.

Not sufficient. Dow to B, C or E.

St2: m is an even integer. Not sufficient. If m = 2 and n = 1 -> m > n but if m = 2 and n = 3 -> m < n.

St 1 + St2: Not sufficient as illustrated by case 1 and case 2.

I think you are making a mistake somewhere !! Well from the different cases you have illustrated it is clear that "A" is sufficient !! Because both cases where m-1,n+1 were consecutive you had m>n therefore statement "1" is clearly sufficient. Hope that helped

Thanks Zatmah! Great catch! Using both the statements, we are down to case 2 only in which m > n.

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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 11:41
1
Donnie84 wrote:
St1: m-1 and n+1 are consecutive positive integers.

Case 1:
m is even, n is odd and m < n
Let m = 2, n = 3
m-1 = 1, n+1 = 4 -> not consecutive positive integers.

Case 2:
m is even, n is odd and m > n
Let m = 2, n = 1
m-1 = 1, n+1 = 2 -> consecutive positive integers.

Case 3:
m is odd, n is even and m < n
Let m = 3, n = 4
m-1 = 2, n+1 = 5 -> not consecutive positive integers.

Case 4:
m is odd, n is even and m > n
Let m = 3, n = 2
m-1 = 2, n+1 = 3 -> consecutive positive integers.

Not sufficient. Dow to B, C or E.

St2: m is an even integer. Not sufficient. If m = 2 and n = 1 -> m > n but if m = 2 and n = 3 -> m < n.

St 1 + St2: Not sufficient as illustrated by case 1 and case 2.

You're almost correct; look at your statement 1 possibilities. Both of the possibilities that show m, n as consecutive integers also have m>n. Therefore statement A is sufficient.
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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 12:12
speedilly wrote:
Donnie84 wrote:
St1: m-1 and n+1 are consecutive positive integers.

Case 1:
m is even, n is odd and m < n
Let m = 2, n = 3
m-1 = 1, n+1 = 4 -> not consecutive positive integers.

Case 2:
m is even, n is odd and m > n
Let m = 2, n = 1
m-1 = 1, n+1 = 2 -> consecutive positive integers.

Case 3:
m is odd, n is even and m < n
Let m = 3, n = 4
m-1 = 2, n+1 = 5 -> not consecutive positive integers.

Case 4:
m is odd, n is even and m > n
Let m = 3, n = 2
m-1 = 2, n+1 = 3 -> consecutive positive integers.

Not sufficient. Dow to B, C or E.

St2: m is an even integer. Not sufficient. If m = 2 and n = 1 -> m > n but if m = 2 and n = 3 -> m < n.

St 1 + St2: Not sufficient as illustrated by case 1 and case 2.

You're almost correct; look at your statement 1 possibilities. Both of the possibilities that show m, n as consecutive integers also have m>n. Therefore statement A is sufficient.

Thanks speedilly! Got it finally. Gosh, this question was really challenging
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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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12 Feb 2014, 23:46
Zatmah wrote:
Donnie84 wrote:
St1: m-1 and n+1 are consecutive positive integers.

Case 1:
m is even, n is odd and m < n
Let m = 2, n = 3
m-1 = 1, n+1 = 4 -> not consecutive positive integers.

Case 2:
m is even, n is odd and m > n
Let m = 2, n = 1
m-1 = 1, n+1 = 2 -> consecutive positive integers.

Case 3:
m is odd, n is even and m < n
Let m = 3, n = 4
m-1 = 2, n+1 = 5 -> not consecutive positive integers.

Case 4:
m is odd, n is even and m > n
Let m = 3, n = 2
m-1 = 2, n+1 = 3 -> consecutive positive integers.

Not sufficient. Dow to B, C or E.

St2: m is an even integer. Not sufficient. If m = 2 and n = 1 -> m > n but if m = 2 and n = 3 -> m < n.

St 1 + St2: Not sufficient as illustrated by case 1 and case 2.

I think you are making a mistake somewhere !! Well from the different cases you have illustrated it is clear that "A" is sufficient !! Because both cases where m-1,n+1 were consecutive you had m>n therefore statement "1" is clearly sufficient. Hope that helped

Indeed, A is sufficient ,:)
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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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17 Feb 2014, 01:48
1
SOLUTION

If m and n are consecutive positive integers, is m greater than n?

(1) m-1 and n+1 are consecutive positive integers --> m>n, if m were less than n than m-1 (integer less than m) and n+1 (integer more than n) wouldn't be consecutive. Sufficient.

Or look at this in another way: stem says that the distance between m and n is 1. Now, if m<n then the distance between m-1 and n+1 would be 3 and they couldn't be consecutive as (1) states. Thus it must be true that m>n.

(2) m is an even integer. Clearly insufficient.

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Re: If m and n are consecutive positive integers, is m greater t  [#permalink]

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23 Feb 2014, 11:46
1
Hi - I have another (slightly different way) that works better and faster for me

If m and n are consecutive positive integers, is m greater than n?

(1) m-1 and n+1 are consecutive positive integers.
For M > N to be consecutive and true,
M = N + 1 and N = M -1
stem (1) give us precisely the algebraic expression we require to prove M > N is true

(2) m is an even integer
Insufficient. For example, if M = 4, N could be 3 or 5.

Hence A alone is sufficient.
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If m and n are consecutive positive integers, is m greater t  [#permalink]

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14 Jun 2015, 02:52
1
Statement 1) Let's pick some consecutive numbers

Case 1
m=15 , n=14
m-1=14, n+1=15

Case2
m=14, n=15
M-1=13 n+1=16 --> not sonsecutive --> m>n (see first case, m&n can be consecutive ONLY in case M>N )

Statement 2) clearly not sufficient
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If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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24 Feb 2016, 02:52
If m and n are consecutive positive integers, is m greater than n ?

(1) m -1 and n + 1 are consecutive positive integers.
(2) m is an even integer.

Arithmetic Properties of numbers
For two integers x and y to be consecutive, it is
both necessary and sufficient that I x - y I = 1.
(1) Given that m-1 and n + 1 are consecutive
integers, it follows that ~ m - 1)- ( n + 1 ~ = 1,
or lm - n - ~ = 1. Therefore, m - n - 2 = 1 or
m - n - 2 = -1. The former equation implies
that m- n = 3, which contradicts the fact
that m and n are consecutive integers.
Therefore, m - n - 2 = -1, or m = n + 1,
and hence m > n; SUFFICIENT.
(2) If m = 2 and n = 1, then m is greater than n.
However, if m = 2 and n = 3, then m is not
greater than n; NOT sufficient

I know all of the topics of the question. But my confusion is that question describes m and n are consecutive positive integers, so why we need
I x - y I = 1. Why we use absolute for positive integers. Would any one like to explain me please.
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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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24 Feb 2016, 02:58
1
If m and n are consecutive positive integers, is m greater than n ?

(1) m -1 and n + 1 are consecutive positive integers.
(2) m is an even integer.

Arithmetic Properties of numbers
For two integers x and y to be consecutive, it is
both necessary and sufficient that I x - y I = 1.
(1) Given that m-1 and n + 1 are consecutive
integers, it follows that ~ m - 1)- ( n + 1 ~ = 1,
or lm - n - ~ = 1. Therefore, m - n - 2 = 1 or
m - n - 2 = -1. The former equation implies
that m- n = 3, which contradicts the fact
that m and n are consecutive integers.
Therefore, m - n - 2 = -1, or m = n + 1,
and hence m > n; SUFFICIENT.
(2) If m = 2 and n = 1, then m is greater than n.
However, if m = 2 and n = 3, then m is not
greater than n; NOT sufficient

I know all of the topics of the question. But my confusion is that question describes m and n are consecutive positive integers, so why we need
I x - y I = 1. Why we use absolute for positive integers. Would any one like to explain me please.

Hi,
we take ABSOLUTE MOD because we are not aware whether x is big or y is big..
if x is bigger, x-y=1..
if y is bigger, x-y=-1..
so |x-y|=1..

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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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24 Feb 2016, 03:07
chetan2u wrote:
If m and n are consecutive positive integers, is m greater than n ?

(1) m -1 and n + 1 are consecutive positive integers.
(2) m is an even integer.

Arithmetic Properties of numbers
For two integers x and y to be consecutive, it is
both necessary and sufficient that I x - y I = 1.
(1) Given that m-1 and n + 1 are consecutive
integers, it follows that ~ m - 1)- ( n + 1 ~ = 1,
or lm - n - ~ = 1. Therefore, m - n - 2 = 1 or
m - n - 2 = -1. The former equation implies
that m- n = 3, which contradicts the fact
that m and n are consecutive integers.
Therefore, m - n - 2 = -1, or m = n + 1,
and hence m > n; SUFFICIENT.
(2) If m = 2 and n = 1, then m is greater than n.
However, if m = 2 and n = 3, then m is not
greater than n; NOT sufficient

I know all of the topics of the question. But my confusion is that question describes m and n are consecutive positive integers, so why we need
I x - y I = 1. Why we use absolute for positive integers. Would any one like to explain me please.

Hi,
we take ABSOLUTE MOD because we are not aware whether x is big or y is big..
if x is bigger, x-y=1..
if y is bigger, x-y=-1..
so |x-y|=1..

thank you I have overcome my confusion. thank you.
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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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24 Feb 2016, 03:16
1
(1) m - 1 and n + 1 are consecutive positive integers.
(2) m is an even integer.

Statement I : Sufficient

Lets take the help of numbers here.

m = 3 and n = 2
m-1 = 2 and n + 1 = 3 ( m-1 and n+1 are consecutive integers).

m = 2 and n = 3
m-1 = 1 and n + 1 = 4 ( m-1 and n+1 are not consecutive integers).

Generalize it,

m = x

n = x + 1

m - 1 = x - 1

n + 1 = x + 2

In this case, m-1 and n+1 are not consecutive positive integers.

Now,

m = x + 1

n = x

m - 1 = x

n + 1 = x + 1

In this case, m-1 and n+1 are consecutive positive integers.

Hence m > n

Statement II: Not Sufficient.

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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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29 Jun 2017, 14:42
If m and n are consecutive positive integers, is m greater than n ?

(1) m-1 and n+1 are consecutive positive integers.

Lets check by substituting some values

$$I - m > n$$

$$m = 4 & n = 3$$

$$m - 1 = 4 - 1 = 3$$

$$n + 1 = 3 + 1 = 4$$

$$I - m > n$$ - Satisfies the Equation as we are getting two consecutive integers 3 & 4 (Also you can check by plugging in other values like 7 & 8)

$$II - m < n$$

$$m = 3 & n = 4$$

$$m - 1 = 3 - 1 = 2$$

$$n + 1 = 4 + 1 = 5$$

$$II - m < n$$ - This does not satisfy the Equation as we are not getting two consecutive equations. (Also you can check by plugging in other values like 8 & 7)

Hence, we see that in order to satisfy the Eq. (1), m has to be greater than n.

Hence (1) =====> is SUFFICIENT

(2) m is an even integer.

m is even, this does not give us any information about n, n can be consecutive integer which is higher than m or consecutive integer which is lower than m.

Hence, (2) =====> is NOT SUFFICIENT

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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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29 Jun 2017, 17:58
I have one small confusion. The question says that "if m and n are consecutive positive integers..." Doesn't this imply automatically that the order is m and then n? How are we unsure that its not m and n rather it could be n and m? Please help me.

Thanks!
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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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29 Jun 2017, 23:52
csaluja wrote:
I have one small confusion. The question says that "if m and n are consecutive positive integers..." Doesn't this imply automatically that the order is m and then n? How are we unsure that its not m and n rather it could be n and m? Please help me.

Thanks!

If it were the way you are saying, we won't have this question. So, x and y are consecutive integers does NOT necessarily mean that y > x.

P.S. Also, note that this is an official question.
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Re: If m and n are consecutive positive integers, is m greater than n ?  [#permalink]

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