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<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

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<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

by Max@Math Revolution » Fri Feb 07, 2020 1:06 am
[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.
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Source: — Data Sufficiency |
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Re: <x> denotes x - 10[x/10] and n is a positive integer. What is the value of <9^n - 1>? ([x] means the greatest intege

by Max@Math Revolution » Sun Feb 09, 2020 5:17 pm
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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
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