If k is an integer and m 0, is 2^k + 3^k = m ?

Question

If k is an integer and m > 0, is 2^k + 3^k = m ?

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

blink005 · Answer

Well, you need to determine whether 2^k + 3^k =m.

1. 4^k + 9 ^k = m^2 - 12
you can simplify it as

2^2k+3^2k+12=m^2
Can you see that this equation can be of the form (a+b)^2=a^2+b^2+2ab when k=1
Otherwise this form can't be simplified any further.
Insufficient

2. k=1
You don't what's m so you can't compare m with a value.
Insufficient

1+2
(4^1+9^1+12)=m^2
m^2=25
m=5
when k=1
2^1+3^1=5=m
Hence sufficient.

Do you follow?

rjkaufman21 · Answer

Oh dang, yes ok I see what you mean now. I'm not great at recognizing the quadratic forms.

Thank you for your help!

Bunuel · Answer

Similar question to practice: if-k-is-an-integer-is-2-k-3-k-m-158354.html

EMPOWERgmatRichC · Answer

Hi All,

While this DS question is a bit "crazy looking", the math behind it is not too difficult. It is perfect for TESTing VALUES.

We're told that K is an integer and that M > 0. We're asked if 2^K + 3^K = M. This is a YES/NO question.

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

IF....
K = 0
4^0 + 9^0 = M^2 - 12
2 = M^2 - 12
14 = M^2
Normally, M would have 2 values, but we were told that M is POSITIVE....
M = root14
The answer to the question (is 2^0 + 3^0 = root14?) is NO.

IF.....
K = 1
4^1 + 9^1 = M^2 - 12
13 = M^2 - 12
25 = M^2
M = 5
The answer to the question (is 2^1 + 3^1 = 5?) is YES.
Fact 1 is INSUFFICIENT

Fact 1: K = 1

This tells us NOTHING about the value of M, so we have no way to know if the answer to the question is YES or NO.
Fact 2 is INSUFFICIENT

Combined, we know that there's just one permissible value for K (1) and the resulting calculation (from Fact 1) yields just one answer: ALWAYS YES.
Combined, SUFFICIENT

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

Madhavi1990 · Answer

Hi!I had a query here: if st 2 says k =1; then we get 2^1+3^1 = m; so m = 5. Why does M have nothing as a value? Why can't we assume LHS = RHS? Would be glad to know my error here.

EMPOWERgmatRichC · Answer

Hi Madhavi1990,

When dealing with DS prompts, you have to be careful to differentiate between information that is given and the question that is ASKED. Here, we're asked IF 2^K + 3^K = M.

Fact 2 tells us that K=1, but it tells us NOTHING about M. 
IF.... M = 5, then the answer to the question is YES.
IF.... M = anything other than 5, then the answer to the question is NO.
Thus, since there's more than one possible answer to the given question, Fact 2 is INSUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

Madhavi1990 · Answer

Thank you, Rich, it was really helpful

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If k is an integer and m > 0, is 2^k + 3^k = m ?

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