## Question

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

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 🙂