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Intern  Joined: 18 Feb 2013
Posts: 29
GMAT 1: 710 Q49 V38 If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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16 00:00

Difficulty:   65% (hard)

Question Stats: 60% (02:17) correct 40% (02:32) wrong based on 389 sessions

### HideShow timer Statistics If m and n are integers and $$\frac{36}{3^4}=\frac{1}{3^m}+\frac{1}{3^n}$$, what is the value of m+n?

A. -2
B. 0
C. 2
D. 3
E. 5

Originally posted by DoItRight on 26 Mar 2013, 14:32.
Last edited by Bunuel on 27 Mar 2013, 04:37, edited 1 time in total.
RENAMED THE TOPIC.
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DoItRight wrote:
If m and n are integers and $$\frac{36}{3^4}=\frac{1}{3^m}+\frac{1}{3^n}$$, what is the value of m+n?

A. -2
B. 0
C. 2
D. 3
E. 5

We have $$\frac{36}{3^4}$$ = $$2^2/3^2$$ = 4/9.

Thus, 4/9 = $$\frac{1}{3^m}+\frac{1}{3^n}$$

or (3+1)/9 = $$\frac{1}{3^m}+\frac{1}{3^n}$$

or 1/3 + 1/9 = $$\frac{1}{3^m}+\frac{1}{3^n}$$

Thus, m = 1, n =2, m+n = 3.

D.
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Not sure why I get this as m+n=2

My workings: 36 / 9 * 9 = 1 / 3^m + 1 / 3^n
4 / 9 = {3^n + 3^m} / 3^(n+m)
since 9 = 3^2, we get 3^2=3^(m+n)
hence m+n = 2

Ans C
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GMAT 1: 640 Q42 V35 ### Show Tags

3
$$\frac{36}{3^4}$$ can be rewritten as $$\frac{36}{81}$$ -> $$\frac{12}{27}$$. $$3^3=27$$. Therefore, m+n=3.

To check $$\frac{1}{3^2}$$ -> $$\frac{3}{27}$$ & $$\frac{1}{3^1}$$ -> $$\frac{9}{27}$$

$$\frac{3}{27}$$ + $$\frac{9}{27}$$ = $$\frac{12}{27}$$
Intern  Joined: 18 Feb 2013
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GMAT 1: 710 Q49 V38 Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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Hey vinaymimani,

How did you know when to break 4/9 into 1/9+3/9? I tried using alegbra like srcc25anu did and got nowhere.
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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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DoItRight wrote:
Hey vinaymimani,

How did you know when to break 4/9 into 1/9+3/9? I tried using alegbra like srcc25anu did and got nowhere.

Firstly, solving it algebraically, we have
$$4/9 = 1/3^m + 1/3^n$$

or 4 = 3^(2-m)+3^(2-n)

Now we know that m,n are integers. Moreover, two expressions, both being powers of 3 add upto 4. Thus, it must be of the form : 3+1. Thus, m=2,n=1 or m=1,n=2.

In either case, m+n = 3.

Now as to how it struck me to split 4/9, when you observe $$4/9 = 1/3^m+1/3^n$$, one might notice that after any cancelling of common factors on the rhs, we should end up with a 9 in the denominator. Also, seeing that 4 = 3+1, one can think that maybe there is a (1/3) and a (1/9) on the rhs, where the lcm is 9, which leads to the (3+1) part in the numerator. I hope it was of some help.
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Intern  Joined: 18 Feb 2013
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GMAT 1: 710 Q49 V38 Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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Quote:
Firstly, solving it algebraically, we have
or 4 = 3^(2-m)+3^(2-n)

When I tried the alegbra way, I got
4 / 9 = 3^n + 3^m / 3^(n+m) I did this but i gave me n+m together but it made the numerator very messy, and hit a dead end.

Quote:
Now as to how it struck me to split 4/9, when you observe , one might notice that after any cancelling of common factors on the rhs, we should end up with a 9 in the denominator. Also, seeing that 4 = 3+1, one can think that maybe there is a (1/3) and a (1/9) on the rhs, where the lcm is 9, which leads to the (3+1) part in the numerator. I hope it was of some help.

Kudos on recognizing the split. Thanks for breaking it down.

Does anyone where I can practice these types of questions? Questions where you have to recognize certain splits?
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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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I tried cross multiplying the right side.
got: 36/3^4 = (3^m)*(3^n)/(3^mn)

from this i wasn't able to find an answer.

Can someone help with this?

How did you guys find the way of breaking it down?
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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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1
ronr34 wrote:
I tried cross multiplying the right side.
got: 36/3^4 = (3^m)*(3^n)/(3^mn)
from this i wasn't able to find an answer.

Can someone help with this?

How did you guys find the way of breaking it down?

You don't cross multiply but take an LCM

For RHS, If you take the LCM, we get

$$(3^n+ 3^m)/ 3^{(m+n)}$$

So, we have 36/81 or 4/9 = $$(3^n+ 3^m)/ 3^{(m+n)}$$

It is better to reduce the LHS to 4/9 and then take 9 to the RHS.

The expression becomes 4= 9/3^m + 9/3^n

or $$4= 3^{(2-m)}+3^{(2-n)}$$

Now when you see a 4 on LHS this should tell you that one of the terms on RHS is of the type 3^0 =1 so either m=2 or n=2,plugging in these values and we can get value for n and m+n=3 in any case

The best way forward is given above by mau5 or you end up pluggin in multiple values(If you don't simplify the algebra) and loose the plot as there will be too many iterations you may have to try before getting the answer.

mau5's method is pretty neat.

Hope it helps
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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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cherryli2015 wrote:
If m and n are integers and 36/(3^4) = 1/(3^m) + 1/(3^n), what is the value of m+n?

Thanks very much ...

hi cherry,
right side = 36/3^4= 4/9..
left side = 1/(3^m) + 1/(3^n)=(3^m+3^n)/3^(m+n) =4/9...
so one can solve for m as 1 when n=2 or m as 2 when n=1.. m+n =3, in each case...
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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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DoItRight wrote:
If m and n are integers and $$\frac{36}{3^4}=\frac{1}{3^m}+\frac{1}{3^n}$$, what is the value of m+n?

A. -2
B. 0
C. 2
D. 3
E. 5

My first thought was to try to write 36 in the form of $$3^x + 3^y$$.

$$3^0=1, 3^1=3, 3^2=9, 3^3=27.$$

We can write 36 = 9 + 27 = $$3^2 + 3^3$$

Using the above in LHS,

$$\frac{36}{3^4}$$ = $$\frac{3^2 + 3^3}{3^4}$$ = $$\frac{3^2}{3^4}$$ + $$\frac{3^3}{3^4}$$ = $$\frac{1}{3^2}$$ + $$\frac{1}{3^1}$$

So m = 2, n = 1 => m + n=3

So D
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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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Easiest way to solve is to simplify the LHS and then test out values for m and n on RHS.

36 = (3^2)(2^2)

36/(3^4) = [(3^2)(2^2)]/(3^4) = (2^2)/(3^2) --> 4/9

m and n combination that will give this is m+n=3 (i.e. 1+2)

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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Hi All,

Quant questions on the GMAT are often built around number 'patterns' of some kind, so your ability to 'play around' with a question and find the pattern(s) behind it can help you to get past complex-looking questions without too much trouble.

Here, we're told that M and N are INTEGERS and that 36/(3^4) = 1/(3^M) + 1/(3^N). We're asked for the value of M+N.

To start, I'm going to do a quick comparison assuming that there were no variables at all...

36/(3^4) = 36/81
1/3 = 27/81

So 1/3 + 1/3 = 54/81 > 36/81. This proves that at least one of the two variables has to be greater than 1 (we have to make at least one of the denominators BIGGER so that we can shrink that fraction and get the sum to equal 36/81.

Let's try making one of the variables 1 and one of the variables 2...

1/3 = 27/81
1/(3^2) = 1/9 = 9/81

27/81 + 9/81 = 36/81
This is an EXACT MATCH for the other side of the equation, so this MUST be the solution. M+N = 1+2 = 3

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Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is  [#permalink]

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The way I solved this was to first minimize the LHS to 4/9. (It makes the numbers easier to play with in your head). Then, I just plugged and played until I found something that works. Which is D. Also note that answer choice A and B have to have at least one negative number somewhere.

Posted from my mobile device Re: If m and n are integers and 36/3^4 = 1/3^m + 1/3^n , what is   [#permalink] 25 Nov 2018, 11:15
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