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If k is an integer, is 2^k + 3^k = m ?

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If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 21 Aug 2013, 11:05
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If k is an integer, is 2^k + 3^k = m ?

(1) 4^k + 9^k = m^2 - 12

(2) k = 1

I don't agree with the OA. If we use statements 1 and 2 together, we get this:

We replace the value of \(k\) in the original question, so the question now is:\(is k = 5 ?\)

Now we replace the value of \(k\) in statement (1), so:

\(m^2 = 25\)

So,\(m = +/- 25\)

There is not indication whether m is possitive.

IMO, the answer: E
[Reveal] Spoiler: OA

Last edited by Bunuel on 21 Aug 2013, 11:26, edited 1 time in total.
Edited the OA.
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Re: If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 21 Aug 2013, 11:16
Expert's post
danzig wrote:
If k is an integer, is 2^k + 3^k = m ?

(1) 4^k + 9^k = m^2 - 12

(2) k = 1

I don't agree with the OA. If we use statements 1 and 2 together, we get this:

We replace the value of \(k\) in the original question, so the question now is: is m = 5 ?

Now we replace the value of \(k\) in statement (1), so:

\(m^2 = 25\)

So, m = +/- 5

There is not indication whether m is possitive.

IMO, the answer: E


Yes, you are right. When we combine we get that m is either 5 or -5. If m=5 then the answer is YES but if m=-5, then the answer is No.

Answer: E.
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Re: If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 30 Aug 2013, 00:26
danzig wrote:
If k is an integer, is 2^k + 3^k = m ?

(1) 4^k + 9^k = m^2 - 12

(2) k = 1

I don't agree with the OA. If we use statements 1 and 2 together, we get this:

We replace the value of \(k\) in the original question, so the question now is:\(is k = 5 ?\)

Now we replace the value of \(k\) in statement (1), so:

\(m^2 = 25\)

So,\(m = +/- 25\)

There is not indication whether m is possitive.

IMO, the answer: E


I don't know what was the original OA but I am assuming it was C, here's how I think the answer should be C, please do correct if there is a flaw in my reasoning.

Given If k is an integer, is \(2^k + 3^k = m\)?
Squaring both sides original equation now becomes \(2^{2k} +3^{2k}+2.6^k=m^2\)

statement 1:\(4^k + 9^k = m^2 - 12\)this can be written as \(2^{2k} +3^{2k}+12 =m^2\)..

Comparing this with original equation we see that this will be equal to original equation only if K=1 since we do not have the value of K, hence insufficient

statement 2 : K=1 , by itself it is insufficient as we do not know the value of M

1+ 2

K=1 then \(m^2\) = 25

this is also what we get from the original equation,doesn't matter what m is, \(m^2\) is 25
Please share your views
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Re: If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 30 Aug 2013, 03:41
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Expert's post
stne wrote:
danzig wrote:
If k is an integer, is 2^k + 3^k = m ?

(1) 4^k + 9^k = m^2 - 12

(2) k = 1

I don't agree with the OA. If we use statements 1 and 2 together, we get this:

We replace the value of \(k\) in the original question, so the question now is:\(is k = 5 ?\)

Now we replace the value of \(k\) in statement (1), so:

\(m^2 = 25\)

So,\(m = +/- 25\)

There is not indication whether m is possitive.

IMO, the answer: E


I don't know what was the original OA but I am assuming it was C, here's how I think the answer should be C, please do correct if there is a flaw in my reasoning.

Given If k is an integer, is \(2^k + 3^k = m\)?
Squaring both sides original equation now becomes \(2^{2k} +3^{2k}+2.6^k=m^2\)

statement 1:\(4^k + 9^k = m^2 - 12\)this can be written as \(2^{2k} +3^{2k}+12 =m^2\)..

Comparing this with original equation we see that this will be equal to original equation only if K=1 since we do not have the value of K, hence insufficient

statement 2 : K=1 , by itself it is insufficient as we do not know the value of M

1+ 2

K=1 then \(m^2\) = 25

this is also what we get from the original equation,doesn't matter what m is, \(m^2\) is 25
Please share your views


It does matter. If m=-5, then the question is: does \(2^k + 3^k = -5\)? And you cannot square this.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 30 Aug 2013, 06:00
Bunuel wrote:
stne wrote:
danzig wrote:
If k is an integer, is 2^k + 3^k = m ?

(1) 4^k + 9^k = m^2 - 12

(2) k = 1

I don't agree with the OA. If we use statements 1 and 2 together, we get this:

We replace the value of \(k\) in the original question, so the question now is:\(is k = 5 ?\)

Now we replace the value of \(k\) in statement (1), so:

\(m^2 = 25\)

So,\(m = +/- 25\)

There is not indication whether m is possitive.

IMO, the answer: E


I don't know what was the original OA but I am assuming it was C, here's how I think the answer should be C, please do correct if there is a flaw in my reasoning.

Given If k is an integer, is \(2^k + 3^k = m\)?
Squaring both sides original equation now becomes \(2^{2k} +3^{2k}+2.6^k=m^2\)

statement 1:\(4^k + 9^k = m^2 - 12\)this can be written as \(2^{2k} +3^{2k}+12 =m^2\)..

Comparing this with original equation we see that this will be equal to original equation only if K=1 since we do not have the value of K, hence insufficient

statement 2 : K=1 , by itself it is insufficient as we do not know the value of M

1+ 2

K=1 then \(m^2\) = 25

this is also what we get from the original equation,doesn't matter what m is, \(m^2\) is 25
Please share your views


It does matter. If m=-5, then the question is: does \(2^k + 3^k = -5\)? And you cannot square this.


Well if you say so, there are many ways algebraic equations can be manipulated and I thought they could be squared .Thank you +1
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Re: If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 25 Dec 2014, 06:40
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Re: If k is an integer, is 2^k + 3^k = m ? [#permalink] New post 25 Dec 2014, 09:58
Expert's post
Hi All,

This question can be solved with a combination of TESTing VALUES and some basic algebra/arithmetic skills.

We're told that K is an integer and we're asked if 2^K + 3^K = M. This is a YES/NO question.

Fact 1: 4^K + 9^K = M^2 - 12

At first glance, I would want to TEST something simple...
K = 1
4^1 + 9^1 = M^2 - 12
4 + 9 = M^2 - 12
25 = M^2
M = +5 or -5

In the question, we have to track BOTH options...
Is 2^1 + 3^1 = 5? The answer is YES
Is 2^1 + 3^1 = -5 The answer is NO
Fact 1 is INSUFFICIENT

Fact 2: K = 1

This tells us NOTHING about M, so the answer COULD be YES or it COULD be NO.
Fact 2 is INSUFFICIENT

Combined, we have the same options that we had in Fact 1 - a YES and a NO.
Combined, INSUFFICIENT

Final Answer:
[Reveal] Spoiler:
E


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Re: If k is an integer, is 2^k + 3^k = m ?   [#permalink] 25 Dec 2014, 09:58
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