t and w are distinct integers between 105 and 100, not inclusive. Which of the following could be the units digit of positive integer ‘n’, if t > w + 1, where ?

0

1

2

3

4

5

6

7

8

9

Answer and Explanation

This week’s question, Big Numbers, seems highly time-consuming, but there is actually a conceptual shortcut. First off, notice the word ‘positive’ integer. What happens, generally speaking, when we have two positive integers, lets just call them x and y, arranged in the following manner, ? Well, depending on which of the variables is larger, can lead to a negative number. While it might be tempting to think that x > y to get a positive number, testing a few numbers shows that this is not the case:

As the numbers become bigger, given that x > y, we are going to get increasingly large negative numbers. Notice how this rule holds true even if we increase the value of y (so that’s it is not always ‘2’)

The point here is not to keep plugging in numbers ad nauseam but to notice a pattern: to make sure we end up with a positive number for , w would have to be greater than t. However, the question states that t > w. So how could we possibly make ‘n’ into a positive number? By making sure that w is an even number, since a negative number taken to an even power will give you a positive number. Therefore, w has to be even, and the only even number in which t > w + 1 holds true is when w = 102. Therefore, t would have to equal 104 (remember, t cannot be greater than 104, because it is between 105 and 100, not inclusive). Arriving at this insight can save a lot of time with plugging in various numbers into the original equation, e.g. 101, 102, etc.

So with w = 102 and t = 104, we have only one possible outcome: . This can be simplified as , which gives us ‘0’, the only possible units digit for n. Answer: A.

Really quick, on that last step, in which I simplified: any time you have a units digit that is 4, when you take it to an even number power, the digit will always be 6. The number 2 as a base follows a pattern of 2, 4, 8, 6, where these numbers represent the units digit. So , where x is a multiple of 4, will always give us 6 as a units digit. Therefore, 6 – 6 = 0. And 0 to any power always ends in 0.

for the above example,why haven’t the values 101 and 103
i.e t=103 and w=101, considered,since these values satisfies both conditions ?
in this case value of n is 2

With each of those sets of values you will end up with a negative number. Remember, any negative number raised to an odd integer will result in a negative integer. The question said that ‘n’ is positive, so we have to take that into account.

What I meant is that if we assume the values are t = 3 and w = 1 we get: (3^1 – 1^3)^1 = 2. The fact that this big number is a positive integer (we don’t have to calculate the exact number ) and ends in ‘2’ means that it could be the value of n.

Magoosh blog comment policy: To create the best experience for our readers, we will only approve comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! 😄 Due to the high volume of comments across all of our blogs, we cannot promise that all comments will receive responses from our instructors.

We highly encourage students to help each other out and respond to other students' comments if you can!

If you are a Premium Magoosh student and would like more personalized service from our instructors, you can use the Help tab on the Magoosh dashboard. Thanks!

for the above example,why haven’t the values 101 and 103

i.e t=103 and w=101, considered,since these values satisfies both conditions ?

in this case value of n is 2

I think there is an inconsistency. In question it says – “if t > w + 1” and explanation says “if t < w + 1"

Thanks for catching that!

Why haven’t the following cases been considered, although they give (+)ve n:

1) when t=4 and w=1. n = 3

2) when t=3 and w=1. n = 2.

Hi Prateek,

With each of those sets of values you will end up with a negative number. Remember, any negative number raised to an odd integer will result in a negative integer. The question said that ‘n’ is positive, so we have to take that into account.

This is definitely a tricky question!

I still don’t understand this explanation.

If t=3 and w=1 and we put those numbers into the formula n=2 (a positive number).

Obviously I’m missing something….

Hi Brien,

Sorry that was confusing :).

What I meant is that if we assume the values are t = 3 and w = 1 we get: (3^1 – 1^3)^1 = 2. The fact that this big number is a positive integer (we don’t have to calculate the exact number ) and ends in ‘2’ means that it could be the value of n.

Hopefully that makes more sense 🙂