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 ?
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.