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# If m and n are positive integers, is m^n < n^m?

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If m and n are positive integers, is m^n < n^m? [#permalink]  22 Sep 2010, 12:37
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If m and n are positive integers, is m^n < n^m?

(1) $$m = \sqrt{n}$$
(2) n > 5

[Reveal] Spoiler:
i disagree with the OA.
My reasoning below
1) if m = 1 & n = 1 then m=sqrt(n) and m^n=n^m
if m=2 & n=4 then m=sqrt(n) and m^n=n^m
if m=3 & n=9 then m=sqrt(n) and m^n>n^m
So if 1) then m^n<n^m always false ==> sufficient

2) we do not know anything about m so insufficient

I assume answer is A. Do you agree ?
[Reveal] Spoiler: OA

Last edited by Bunuel on 15 Jul 2013, 22:20, edited 2 times in total.
Edited the OA.
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Re: Exponents-roots DS [#permalink]  22 Sep 2010, 13:08
tatane90 wrote:
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

i disagree with the OA.
My reasoning below
1) if m = 1 & n = 1 then m=sqrt(n) and m^n=n^m
if m=2 & n=4 then m=sqrt(n) and m^n=n^m
if m=3 & n=9 then m=sqrt(n) and m^n>n^m
So if 1) then m^n<n^m always false ==> sufficient

2) we do not know anything about m so insufficient

I assume answer is A. Do you agree ?

(1) : $$m^n < n^m$$
$$m^{m^2} < n^m$$
$$(m^2)^{m^2/2} < n^m$$
$$n^{m^2/2} < n^m$$

For n,m integers and both greater than 1, this implies

$$m^2/2 < m$$
$$m(m-2) < 0$$

This expression is false for all m>2

Also we know for m=1 and m=2 that m^n=n^m ... so again the expression is false

So (1) is sufficient

(2) : Not sufficient as only condition on n

So I agree answer is A

The answer would be (c) if the original question was m^n <= n^m
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Re: Exponents-roots DS [#permalink]  22 Sep 2010, 13:35
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

from 1

m = sqrt n ie: n = m^2

is m^(m^2)< m^2m , is m^m(m-2) < 1 is m(m-2)<0 is m>2.... insuff

from 2

obviously insuff

both suff

n>5, n = m^2 ( try worst case scenario n = 9 ) thus m = 3 >2...suff

Last edited by yezz on 22 Sep 2010, 13:43, edited 1 time in total.
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Re: Exponents-roots DS [#permalink]  22 Sep 2010, 13:43
yezz wrote:
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

from 1

m = sqrt n ie: n = m^2

is m^(m^2)< (m^2)^m

plug m = 1 or 2 (no), plug m = 7 (yes) insuff

from 2

obviously insuff

both suff

C

With m=7, LHS is 7^49 or 49^(24.5)
RHS is 49^(7)
So LHS > RHS
So answer is NO not YES

A is sufficient !
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Re: Exponents-roots DS [#permalink]  22 Sep 2010, 13:44
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Expert's post
tatane90 wrote:
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

i disagree with the OA.
My reasoning below
1) if m = 1 & n = 1 then m=sqrt(n) and m^n=n^m
if m=2 & n=4 then m=sqrt(n) and m^n=n^m
if m=3 & n=9 then m=sqrt(n) and m^n>n^m
So if 1) then m^n<n^m always false ==> sufficient

2) we do not know anything about m so insufficient

I assume answer is A. Do you agree ?

It seems that you are right, answer should be A.

Answer to be C question shouldn't say that $$m$$ and $$n$$ are integers. In this case if $$m=\sqrt{n}=\sqrt{2}$$ then $$m^n=\sqrt{2}^2=2<2^{\sqrt{2}}=n^m$$, so (1) wouldn't be sufficient.

Also the question would be a little bit trickier in this case.

yezz wrote:
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

from 1

m = sqrt n ie: n = m^2

is m^(m^2)< (m^2)^m

plug m = 1 or 2 (no), plug m = 7 (yes) insuff
from 2

obviously insuff

both suff

C

If $$m=7$$ then $$n=49$$ and $$7^{49}>49^7$$, so answer is still NO.
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Re: Exponents-roots DS [#permalink]  22 Sep 2010, 15:28
I will go with C...OA seems correct

1. With one alone I get

Is $$m^n > n^m$$
Is $$(\sqrt{n})^n$$ >$$n^(\sqrt{n}$$
Is $$n^(\frac{n}{2}) = n^(\sqrt{n})$$. Since I made bases same I have to answer is n/2 greater than squreroot n....cannot answer

2. knowing n alone will not tell me about expression

If I combine both I can answer my question - Is N/2 greater than $$\sqrt{n}$$
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Re: Exponents-roots DS [#permalink]  22 Sep 2010, 15:39
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I think I answered the question by reversing the signs of equality. Is the question really correct?
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Re: Exponents-roots DS [#permalink]  12 Sep 2012, 15:28
Bunuel wrote:
tatane90 wrote:
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

i disagree with the OA.
My reasoning below
1) if m = 1 & n = 1 then m=sqrt(n) and m^n=n^m
if m=2 & n=4 then m=sqrt(n) and m^n=n^m
if m=3 & n=9 then m=sqrt(n) and m^n>n^m
So if 1) then m^n<n^m always false ==> sufficient

2) we do not know anything about m so insufficient

I assume answer is A. Do you agree ?

It seems that you are right, answer should be A.

Answer to be C question shouldn't say that $$m$$ and $$n$$ are integers. In this case if $$m=\sqrt{n}=\sqrt{2}$$ then $$m^n=\sqrt{2}^2=2<2^{\sqrt{2}}=n^m$$, so (1) wouldn't be sufficient.

Also the question would be a little bit trickier in this case.

yezz wrote:
If m and n are positive integers, is m^n < n^m?
(1) m = sqrt(n)
(2) n > 5

from 1

m = sqrt n ie: n = m^2

is m^(m^2)< (m^2)^m

plug m = 1 or 2 (no), plug m = 7 (yes) insuff
from 2

obviously insuff

both suff

C

If $$m=7$$ then $$n=49$$ and $$7^{49}>49^7$$, so answer is still NO.

Hi Bunuel,

Can you discuss this question from scratch as a new question. As per me answer is A.
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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  13 Sep 2012, 03:56
Expert's post
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If m and n are positive integers, is m^n < n^m?

(1) m = sqrt(n) --> $$m^2=n$$. Substitute $$n$$ in the question: is $$m^{m^2}<(m^2)^m$$? --> is $$m^{m^2}<m^{2m}$$? Now, if $$m$$ is 1 or 2, then $$m^{m^2}=m^{2m}$$, so the answer is NO and if $$m$$ is an integer greater than 2, then $$m^{m^2}>m^{2m}$$, so the answer is still NO. Sufficient.

(2) n > 5. If $$m=1$$, then the answer is YES but if $$m=2$$, then the answer is NO. Not sufficient.

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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  13 Sep 2012, 05:50
Bunuel wrote:
If m and n are positive integers, is m^n < n^m?

(1) m = sqrt(n) --> $$m^2=n$$. Substitute $$n$$ in the question: is $$m^{m^2}<(m^2)^m$$? --> is $$m^{m^2}<m^{2m}$$? Now, if $$m$$ is 1 or 2, then $$m^{m^2}=m^{2m}$$, so the answer is NO and if $$m$$ is an integer greater than 2, then $$m^{m^2}>m^{2m}$$, so the answer is still NO. Sufficient.

(2) n > 5. If $$m=1$$, then the answer is YES but if $$m=2$$, then the answer is NO. Not sufficient.

Thanks Bunuel for the quick reply
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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  15 Jul 2013, 22:22
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Bumping for review and further discussion.
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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  16 Jan 2015, 07:58
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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  16 Jan 2015, 11:52
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Hi All,

This DS question can be solved by TESTing VALUES.

We're told that M and N are POSITIVE INTEGERS. We're asked if M^N < N^M. This is a YES/NO question.

Fact 1: M = \sqrt{N}

IF....
N = 1
M = 1
1^1 is NOT < 1^1 and the answer to the question is NO.

N = 4
M = 2
2^4 is NOT < 4^2 and the answer to the question is NO.

N = 9
M = 3
3^9 is NOT < 9^3 and the answer to the question is NO.
This pattern will continue; the answer to the question is ALWAYS NO.
Fact 1 is SUFFICIENT.

Fact 2: N > 5

This tells us NOTHING about the value of M, so this is probably insufficient. Here's the proof.

IF...
M = 1
N = 6
1^6 < 6^1 and the answer to the question is YES.

IF....
M = 6
N = 6
6^6 is NOT < 6^6 and the answer to the question is NO.
Fact 2 is INSUFFICIENT

[Reveal] Spoiler:
A

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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  25 Jan 2015, 14:34
Very good question indeed for practice.

its given M= sqrt(N)

to show is whether M^n < N^m
substitute N = M^2 in the above equation we get

M^n < M^2m

it means n< 2m (As bases are same)

which means n < 2sqt(n)

above equation can be reduced to sqrt(n) < 1.

Sqrt(n) can never be less than 1 as n is positive integer.

So using stmt 1 we can derive the answer as NO.

Stmnt2 does not have any reference of M. so not sufficient.

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If m and n are positive integers, is m^n < n^m? [#permalink]  09 Jun 2015, 22:06
It would have been interesting to see if the inequality had been reversed. => m^n > n^m

In this case.. there is a special value which will fail it... m = 2 & n = 4 ... i.e. .. m = n^1/2 & 2^4 = 4^2... In this case it doesn't satisfy the inequality but all other values do.. So then answer would have been C to ensure that m =! 2.
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If m and n are positive integers, is m^n < n^m? [#permalink]  26 Jun 2015, 21:09
Bunuel wrote:
If m and n are positive integers, is m^n < n^m?

(1) m = sqrt(n) --> $$m^2=n$$. Substitute $$n$$ in the question: is $$m^{m^2}<(m^2)^m$$? --> is $$m^{m^2}<m^{2m}$$? Now, if $$m$$ is 1 or 2, then $$m^{m^2}=m^{2m}$$, so the answer is NO and if $$m$$ is an integer greater than 2, then $$m^{m^2}>m^{2m}$$, so the answer is still NO. Sufficient.

(2) n > 5. If $$m=1$$, then the answer is YES but if $$m=2$$, then the answer is NO. Not sufficient.

if I compare with base using "n" rather than "m"

I get
following equation after reduction :

m^n < n^m reduces to

n^(n/2) < (n)^sqrt (n)

means we need find if
n/2 < Sqrt (n), Given n is perfect square bcs m = sqrt(n) and m is integer.

for n=1 : yes
for n=4 : no
for n=64 : No
for n=9 : No

then how can we answer : A is sufficient.

on other side if n>5 then
Answer B: is always hold. So sufficient....

I am sure I must be wrong somewhere in logic!!!
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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  26 Jun 2015, 23:19
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Jam2014 wrote:
Bunuel wrote:
If m and n are positive integers, is m^n < n^m?

(1) m = sqrt(n) --> $$m^2=n$$. Substitute $$n$$ in the question: is $$m^{m^2}<(m^2)^m$$? --> is $$m^{m^2}<m^{2m}$$? Now, if $$m$$ is 1 or 2, then $$m^{m^2}=m^{2m}$$, so the answer is NO and if $$m$$ is an integer greater than 2, then $$m^{m^2}>m^{2m}$$, so the answer is still NO. Sufficient.

(2) n > 5. If $$m=1$$, then the answer is YES but if $$m=2$$, then the answer is NO. Not sufficient.

if I compare with base using "n" rather than "m"

I get
following equation after reduction :

m^n < n^m reduces to

n^(n/2) < (n)^sqrt (n)

means we need find if
n/2 < Sqrt (n), Given n is perfect square bcs m = sqrt(n) and m is integer.

for n=1 : yes
for n=4 : no
for n=64 : No
for n=9 : No

then how can we answer : A is sufficient.

on other side if n>5 then
Answer B: is always hold. So sufficient....

I am sure I must be wrong somewhere in logic!!!

Look at the highlighted steps only

n^(n/2) < (n)^sqrt (n)

for n=1 : yes The answer is not Yes it's No here as well because n^(n/2) will NOT be less than (n)^sqrt (n) for n=1

Statement 2: n>5

@n=6, and m=1, m^n will be less than n^m
@n=6, and m=2, m^n will NOT be less than n^m
NOT SUFFICIENT

I hope it helps!
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Re: If m and n are positive integers, is m^n < n^m? [#permalink]  27 Jun 2015, 02:52
GMATinsight wrote:
Jam2014 wrote:
Bunuel wrote:
If m and n are positive integers, is m^n < n^m?

(1) m = sqrt(n) --> $$m^2=n$$. Substitute $$n$$ in the question: is $$m^{m^2}<(m^2)^m$$? --> is $$m^{m^2}<m^{2m}$$? Now, if $$m$$ is 1 or 2, then $$m^{m^2}=m^{2m}$$, so the answer is NO and if $$m$$ is an integer greater than 2, then $$m^{m^2}>m^{2m}$$, so the answer is still NO. Sufficient.

(2) n > 5. If $$m=1$$, then the answer is YES but if $$m=2$$, then the answer is NO. Not sufficient.

if I compare with base using "n" rather than "m"

I get
following equation after reduction :

m^n < n^m reduces to

n^(n/2) < (n)^sqrt (n)

means we need find if
n/2 < Sqrt (n), Given n is perfect square bcs m = sqrt(n) and m is integer.

for n=1 : yes
for n=4 : no
for n=64 : No
for n=9 : No

then how can we answer : A is sufficient.

on other side if n>5 then
Answer B: is always hold. So sufficient....

I am sure I must be wrong somewhere in logic!!!

Look at the highlighted steps only

n^(n/2) < (n)^sqrt (n)

for n=1 : yes The answer is not Yes it's No here as well because n^(n/2) will NOT be less than (n)^sqrt (n) for n=1

Statement 2: n>5

@n=6, and m=1, m^n will be less than n^m
@n=6, and m=2, m^n will NOT be less than n^m
NOT SUFFICIENT

I hope it helps!

My take away is that I can't compare just power. Based on what Bases are, results can be different!!!!
Great concept and question.
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If m and n are positive integers, is m^n < n^m? [#permalink]  30 Jun 2015, 22:41
Expert's post
If m and n are positive integers, is m^n < n^m?

(1) m=√n
Squaring both sides yields m^2 = n
This can be substituted into the original equation m^(m^2) < (m^2)^m
This is sufficient (no). You can try one or two numbers to confirm.

(2) n > 5 Since we know nothing of m, this is not sufficient.

A
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If m and n are positive integers, is m^n < n^m?   [#permalink] 30 Jun 2015, 22:41
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