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GMAT Practice Question (One of Hundreds)
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}?
Correct Answer: B
An odd number is never divisible by an even number. What value of z could make the expression (2,066)^{z} equal to an odd number that is a factor of any number?
An odd integer (e.g., 3,176,793) is not divisible by an even integer. For example, 3 is not divisible by 2 nor is 15 divisible by 2.
(3,176,793/even integer) --> non integer
The only way (2,066)^{z} can possibly be a divisor of 3,176,793 is if (2,066)^{z} is an odd number. However, if z is any positive integer, (2,066)^{z} will be an even number. (More specifically, it will have a units digit of 6). As a result, (2,066)^{z} will not be a factor of 3,176,793 if z is a positive integer.
Since the question explicitly says that (2,066)^{z} is a factor of 3,176,793, you know that, somehow, (2,066)^{z} must be an odd number.
Remember that any number raised to the power of 0 will be 1.
(any real number)^{0} = 1
The key to this problem is realizing that if z = 0, which is allowed because the question stem said z "is a non-negative number," (2,066)^{z} will equal 1. Since the only way z can be a factor 3,176,793 is if z = 0, you know that z = 0.
(3,176,793/(2,066)^{0}) is the only way 2,066^{z} is a factor of 3,176,793
You can now rewrite the question as follows:
0^{(any positive integer)} – (any positive integer)^{0}
z^{x} – x^{z} = 0^{x} – x^{0} = 0^{(pos int)} – (pos int)^{0}
Since 0 raised to any positive integer equals 0 and any positive integer raised to 0 is 1, the question boils down to: 0 – 1 = -1.
0^{(pos int)} – (pos int)^{0} = 0 – 1 = -1