If m and n are negative integers what is the value of m*n

Question

(1) m^n = 1/81
(2) n^m = -(1/64)

Bunuel · Answer

There is something wrong with this question as statements contradict each other: (1) says m^n=1/81 and (2) says m^n=-(1/64). Guess (2) should read: n^m=-(1/64).

If m and n are negative inegers what is the value of m*n?

(1) m^n=1/81 --> as both m and n are negative integers then  m^n=\frac{1}{81}=(-9)^{-2}=(-3)^{-4}  -->  mn=18  or  mn=12  (note that as negative integer in negative integer power gives positive number then the power must be negative even number). Not sufficient.

(2) n^m=-(1/64) --> as the result is negative then  m  must be negative odd number -->  n^m=-\frac{1}{64}=(-4)^{-3}=(-64)^{-1}  -->  mn=12  or  mn=64 . Not sufficient.

(1)+(2) Only one pair of negative integers  m  and  n  satisfies both statements  m=-3  and  n=-4  -->  mn=12 . Sufficient.

Answer: C.

Bunuel · Answer

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jakelong1 · Answer

Hi Bunuel,

Can you please explain how did you come to the conclusion that it should be n^m and not m ^ n in the second sentence.

Bunuel · Answer

Sure.

On the GMAT, two data sufficiency statements always provide TRUE information and these statements never contradict each other.

Now, if the statements were:
 (1) m^n =  1/81 
(2) m^n =  -1/64  
Then they would clearly contradict each other, thus I assumed that one of them must be different.

Does this make sense?

medic19 · Answer

The hunch is that whenever the GMAT sets out 2 formulae that are visually similar then testing cases will probably work and certain numbers will be in common in S1 and S2. Not sure, if this is the most time efficient way but I was a bit biased since I saw this is a 700 question and was therefore a bit skeptical that this is an easy question.

S1. Set out all the multiples you can think of and neatly set out the applicable values of M (9,3) and N (2,4). Obviously not sufficient as 9x2 and 4x3 produce different results

S2. Take same approach as above except there are more values to think of for each: M (2,3,6), N(8,4,2).

You can also think backwards: what is the square root of each number (9 of 81, 8 of 64) and are there any factors of that square root (3 for 9) and (4 and 2 for 8).

smmc29 · Answer

(1) m^n = (1/81)

Since both m,n < 0

n should be even.

81 can be 9^2 or 3^4

If n = -2, then m=-9 -> mn = 18
If n = -4, then m=-3 -> mn = 12

INSUFFICIENT

(2) n^m = -(1/64)

Since the expression is negative, m should be odd.

64 = 64^1 or 4^3

If m=-1, then n=-64 -> mn = 64
If m=-3, then n=-4 -> mn=12

INSUFFICIENT

(1)(2) the only common case for the two is when m=-3, and n=-4 -> mn=12

SUFFICIENT

The answer is C.

If m and n are negative integers what is the value of m*n

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