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Algebra; if m and n are integers

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Algebra; if m and n are integers

by lucas211 » Fri Apr 08, 2016 1:45 am
Hello BTG

Would appreciate some help to the following question:

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

a) -2
b) 0
c) 2
d) 3
e) 5

Thanks in advance
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by MartyMurray » Fri Apr 08, 2016 3:37 am
If you are not sure how to handle a question, you could start by just simplifying something to get some insight.

In this case, you could start by reducing 36/3� to 4/3².

4/3² = 1/3² + 3/3² The right side of that expression is starting to look like 1/3� + 1/3�.

I'll let you take it from here.
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by lucas211 » Fri Apr 08, 2016 4:44 am
@ Marty Murray

Hi Marty

So on the right side we multiply each term by 1 ( (3^n/3^n) in the first term and (3^m/3^m)) to get:

3^n/3^m+n + 3^m/3^m+n = (3^n +3^m)/ 3^m+n. so;

4/3^2 = (3^n + 3^m) / 3^m+n -which means that m+n equals 2 ?

Is this approach right?

Thanks in advance
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by MartyMurray » Fri Apr 08, 2016 5:41 am
What makes you sure that (3^n + 3^m) = 4?

You need to get to the point where the numerators on the right side of 4/3² = 1/3² + 3/3² are both 1.

What could you do to make the numerators of both 1/3² and 3/3² 1?

I see that the numerator of 1/3² is already 1.
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by [email protected] » Fri Apr 08, 2016 9:24 am
Hi lucas211,

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

Final Answer: D

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by lucas211 » Sat Apr 09, 2016 1:20 am
Marty Murray wrote:What makes you sure that (3^n + 3^m) = 4?

You need to get to the point where the numerators on the right side of 4/3² = 1/3² + 3/3² are both 1.

What could you do to make the numerators of both 1/3² and 3/3² 1?

I see that the numerator of 1/3² is already 1.

Hi Marty
Thanks for your help so far. This is one of places where I still struggle.
I can see, that we can't be sure about that (3^n + 3^m) equals 4.

To make the numerator in 3/3^2 "1" - we can divide both numerator and denominator by 3, resulting in 1/3, but then we have removed the square in the denominator at the same time ?

Thanks for your patience - I seem to be a bit lost in this question.

Lucas
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by MartyMurray » Sat Apr 09, 2016 4:52 am
Haha Lucas. You got the answer, and you don't see it.

Look at what you created and look at the question.
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by lucas211 » Sun Apr 10, 2016 5:24 am
Marty Murray wrote:Haha Lucas. You got the answer, and you don't see it.

Look at what you created and look at the question.
Hi Marty

Ahh - got it!

36/3^4 = 1/3^m + 1/3^n

36/3^4 = 4/9 => 4/3^2 or 1/3^2 (which fits one of the terms on the right side) + 3/3^2 (which divided by 3 will be 1/3^1 and fit the other of the terms on the right side).

1/3^2 + 3/3^2 = 1/3^2 + 1/3^1

since the bases are the same, we can just add the exponents = 3.

So "d" is the answer.

Thanks a lot lot - it is truly appreciated!

Lucas
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by MartyMurray » Sun Apr 10, 2016 7:18 am
lucas211 wrote:1/3^2 + 3/3^2 = 1/3^2 + 1/3^1

since the bases are the same, we can just add the exponents = 3.

So "d" is the answer.

Thanks a lot lot - it is truly appreciated!

Lucas
Yessssssss

Sure thing.
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by Matt@VeritasPrep » Mon Apr 11, 2016 1:12 pm
Plugging in integers is certainly easier, but we could also swing this algebraically.

(1/3)� + (1/3)� = 4/9

(3� + 3�) / (3�+�) = 4/9

9*(3� + 3�) = 4*(3�+�)

3�+² + 3�+² = (3+1)*(3�+�)

3�+² + 3�+² = 3�+�+¹ + 3�+�

So two solutions are

m+2 = m+n+1 and n+2 = m+n

or

m+2 = m+n and n+2 = m+n+1

In the first case, n = 1 and m = 2, and in the second case, m = 1 and n = 2. Either way, m + n = 3.
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