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Magoosh Brain Twister: Big Numbers – Explanation



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 n =  (t^w - w^t)^w?

  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
  6. 5
  7. 6
  8. 7
  9. 8
  10. 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, x^y - y^x? Well, depending on which of the variables is larger,x^y - y^x 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:

3^2 - 2^3 = 1

4^2 - 2^4 = 0

5^2 - 2^5 = -7

6^2 - 2^6 =  -28

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’)

4^3 - 3^4 = -17

5^3 - 3^5 = some even more negative number

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 t^w - w^t, 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: (104^102 - 102^104)^102. This can be simplified as (4^2 - 2^4)^even, 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 2^x, 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.


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7 Responses to Magoosh Brain Twister: Big Numbers – Explanation

  1. Shruti August 5, 2015 at 4:45 pm #

    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

  2. Prateek Sharma February 10, 2015 at 6:59 pm #

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

    • Chris Lele
      Chris Lele February 11, 2015 at 11:30 am #

      Thanks for catching that!

  3. Prateek Sharma February 10, 2015 at 6:54 pm #

    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.

    • Chris Lele
      Chris Lele February 11, 2015 at 11:34 am #

      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!

      • brien May 10, 2015 at 3:33 pm #

        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….

        • Chris Lele
          Chris Lele May 14, 2015 at 5:00 pm #

          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 🙂

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