<x> denotes x - 10[x/10] and n is a positive integer. What is the value of <9^n - 1>? ([x] means the greatest integer l

[Max@Math Revolution]

[GMAT math practice question]

<x> denotes x - 10[x/10] and n is a positive integer. What is the value of <9^n - 1>? ([x] means the greatest integer less than or equal to x.)

1) <9^n - 1> is not positive.
2) n is an even number.

[Max@Math Revolution]

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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

<x> means the unit digit of x.
For example, if x = 123, then x - 10[x/10] = 123 - 10[12.3] = 123 - 120 = 3.

We have 9^1 = 9, 9^2 = 81, 9^3 = 729, 9^4 = 6561, ...
Then, 9^1 - 1 = 8, 9^2 - 1 = 80, 9^3 - 1 = 728, 9^4 - 1 = 6560, ....
We notice that if n is an odd number, the unit digit of 9^n - 1 is 8, and if n is an even number, the unit digit of 9^n - 1 is 0.

The question asks what the unit digit of 9^n - 1 is.
Condition 2) tells us that n is an even number. Therefore 9^n - 1 is 0. Since condition 2) yields a unique solution, it is sufficient.

Condition 1)
Since the only possible values of <9^n - 1> are 0 and 8, <9^n - 1> is 0 if <9^n - 1> is not positive.

Since condition 1) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D