If x is a positive integer and z is a non-negative integer such that

Question

If x is a positive integer and z is a non-negative integer such that (2,066)^z is a divisor of 3,176,793 what is the value of z^x - x^z?

A. -81
B. -1
C. 0
D. 1
E. It Cannot Be Determined

AKProdigy87 · Accepted Answer

The answer is  B: -1 .

(2,066)^z  is given to be a divisor of 3,176,793. We also know that z is non-negative integer. Just doing a quick check on possible values of z:

(2,066)^2  >  (2,000)^2  =  4,000,000  >  3,176,793

So we can conclude that z = 0 or 1. However, 2066^1 = 2066 is an even number, and 3,176,793 is odd; an even number obviously can't be the divisor of an odd number. So we can rule out z = 1 as well. Therefore, z = 0.

z^x - x^z

= 0^{(positive integer)} - (positive integer)^0

= 0 - 1

= -1

Bunuel · Answer

If x is a positive integer and z is a non-negative integer such that 2,066^z is a divisor of 3,176,793, what is the value of z^x - x^z?

A. -81
B. -1
C. 0
D. 1
E. It Cannot Be Determined

3,176,793 is an odd number. The only way it to be a multiple of 2,066^z (even number in integer power) is when \(z=0\), in this case \(2,066^z=2,066^0=1\) and 1 is a factor of every integer. Hence \(z=0\) --> \(z^x-x^z=0^x-x^0=0-1=-1\).

Answer: B.

Must know for the GMAT: \(a^0=1\), for \(a\neq{0}\) - any nonzero number to the power of 0 is 1. Important note: the case of 0^0 is not tested on the GMAT.

Similar questions to practice:
https://gmatclub.com/forum/tough-and-tri ... l#p1029223
https://gmatclub.com/forum/if-n-is-a-non ... 01264.html

Hope it helps.

KarishmaB · Answer

Let me point out one thing here: When the question maker takes the pains to type 'z is a non-negative integer' rather than 'z is a positive integer', it means z = 0 is a decision point. Nothing a question maker writes is without reason hence pay special attention to this. First put z = 0 and you see that (2066)^0 = 1 which is a divisor of every number.

Now, it may not occur to you immediately that the huge number is odd wile 2066 is even (which means no positive integer power of 2066 can be a divisor of the huge number). The second clue here is the huge number itself. It is not possible that GMAT expects me to divide 3,176,793 by 2066 or multiply 2066 with another 2066. The big numbers are just there to make your heart race and make you even more nervous and sweaty. The bigger the numbers in the question, the bigger should be your smile. You definitely won't have to calculate anything. Just think of your simple even/odd, positive/negative etc concepts and you should have your answer in seconds.

KarishmaB · Answer

A quick tip: If the question says "non-negative integer", alarms should go off in your head and you should immediately consider the case of 0. It is easier to say positive integers. When the test maker goes for "non negative integer", it usually means that 0 will have some role to play in the solution. Also considering the huge numbers the question deals with, it is a good idea to try 0 and 1 (if needed). 
The moment you put z = 0, you get the answer as -1. Since it is a PS question, you are done.

ScottTargetTestPrep · Answer

Since 2,066 is even and 3,176,793 is odd, we see that the only possible value for z is 0. In other words, 2,066 does not evenly divide 3,176,793, and no multiple of 2,066 will evenly divide 3,176,793, either. Thus, z must equal 0, making 2,066^0 = 1 (and we see that 1 does evenly divide into 3,176,793).

Since z = 0, then z^x = 0^x = 0, and x^z = x^0 = 1. Thus, their difference is 0 - 1 = -1.

Answer: B

exc4libur · Answer

Rules:
e/e=even,odd,proper_fraction,na
o/o=odd,p_fraction
e/o=even,p_fraction
o/e=p_fraction,not-defined

3,176,793=odd
2066=even

2066 is not factor of 3,176,793: odd/even=not-integer

2066^z is factor of 3,176,793 only if z=0

z^x - x^z = 0^x-x^0 = 0-1 = -1

Ans (B)

bumpbot · Answer

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If x is a positive integer and z is a non-negative integer such that

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