If k^(xy)=1, where k#0, which of the following must be true

Question

If  k^{xy}=1 , where  k
eq0 , which of the following must be true?

I. k=1 
II. x=0 
III. y=0

A. I only
B. II only
C. III only
D. I and II
E. None

Bunuel · Accepted Answer

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions  if you can prove that a statement is NOT true  for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

k^{xy}=1  --> 3 cases:

xy=0  and  k=any \ integer ;
 k=1  and  xy=any \ integer ;
 k=-1  and  xy=even .

So if we take for example the third case:  k=-1  and  xy=2  then none of option MUST be true.

Answer: E.

subhajeet · Answer

Bunnel if we take all the 3 statements together then can't that be an answer??

Bunuel · Answer

If k=1, x=0 and y=0 then k^(xy)=1^0=1, so this is certainly one of the possibilities (though not the only one: refer to my solution above). Also notice that we are asked which of the following MUST be true not COULD be true.

Hope it's clear.

subhajeet · Answer

Bunnel thanks for the explanation. Got confused between 'could be true' and 'must be true'

Joy111 · Answer

a simpler way:

option 1) k=1 , not necessarily must be true as K can be 2,3,34, etc any number as long as either x or y is equal to 0 ( zero) ,

option 2)x=0 , doesn't have to be true because y = 0, and any number raised to 0 equals 1.so condition holds even if x is not equal to 0

option 3) y=0, doesn't have to be true as x=0 and the condition could still hold .

so we can see none of the options MUST be true , for the condition to hold .

Although Bunuel's solution is better a usual.

laddaboy · Answer

which of the following must be true? - This is a a very important part.

1) k = 1
Then definitely 1 ^ xy is always 1, but say k = 2 and xy = 0 even now K^xy = 1 . Hence K can actually take any integer other than 1. So k = 1 must be true is false, since K can also be 2,3,4 etc...

Following a similar logic the best option is E

If k^(xy)=1, where k#0, which of the following must be true

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If k^(xy)=1, where k#0, which of the following must be true

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