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Units Digit Questions
The units digit question. This is a very frequent question type on the test. Almost every quant section will have at least one of these. And it looks really hard. People who are not familiar with this question type, they see it, they panic because it looks impossibly difficult.
But it’s actually very easy. So for example, a typical question of this sort, what is the units digit of 57 to the power of 123? Now your first impulse might be, oh my, I need a calculator to figure out what 57 to the power of 123 is. Well, I’ll share with you something, this is not something you’d be able to figure out on your own necessarily, but I’ll share.
As it happens, this number is 216 digits long. So in other words, no calculator in the world is gonna be able to figure out 57 to the power of 123 for you. It turns out there are web applications, for example if you’re familiar with Wolfram Alpha. That’s something where you could go, you could enter this, you could actually see all 216 digits if you are really fascinate with that thing.
But the point is no calculator gonna help you with this. Now, at this point this probably really goes your mind. Why is the test ask me to do something that no calculator on the planet can do? Well lets be careful here. No calculator on the planet could figure out that entire power, all 216 digits, but we don’t need all 216 digits, we only need the units digit.
And it turns out, that is a remarkably easy question. So first of all, I’ll point out the general strategy that we’re going to use. We’re just gonna look for a repeating pattern. And then we’re going to figure out where that pattern will be at the desired power. So, I’ll have to explain this in more detail. But first of all, we need, a really big mathematical idea.
Here’s the big mathematical idea. The units digit of any product will be influenced only by the units digits of the two factors. Therefore, we only need to consider single digit products when tackling a units digit question. Now this is a profound idea, and it takes some, some time to sink in.
Let’s think about this for a second. 3 times 6 we know is 18. So, we multiply unit digit of a 3 times the unit digit of 6, I get a units digit of an 8. That means any long number ending in 3 times any long number ending in 6 will be a long number ending in 8.
In other words, all that matters is the unit digits. The unit digits of the two factors is the only thing that determines the units digit of a product. Now if this is a hard idea to understand, what I strongly suggest is stop this video, pick up the calculator, and actually try this for yourself. Enter some numbers.
Say, enter a number in the hundreds that has a unit digit of three. Enter another number in the hundreds that has a unit digit of six. Multiply them together. Time and time again, you will get some bigger number with a units digit of eight. You’ll always get that same units digit in the answer. Then try some other combinations of digits.
Units digits of two, unit digit of seven, unit digit of three, unit digit of nine. Something like that. Just pick your own combinations. Do the single digit multiplication. So, 3 times 9 for example is 27, so unit digit of 3 times a unit digit of 9 will always have a units digit of 7.
Again, experiment with this and verify for yourself on your calculator that it works. If your not a math-y sort of person. One way that you build intuition for numbers is actually playing with numbers, and seeing these patterns. That’s why it’s very important actually to pick up a calculator and verify for yourself that this pattern works.
This is the pattern we’re going to employ. So, first of all, our big grand dad question here. 57 to the power of 123. Well first of all, the tens digit doesn’t matter at all, so I might as well just consider powers of seven, because any number that ends in seven, when I take powers of it, is gonna have the same units digit as just the powers of seven.
Only the units digit matters. So 7 to the 1st of course just 7. 7 squared, 49. I’m just gonna write that as something 9. We have a units digit of 9. 9 and then multiply by 7 again.
9 times 7, 63. So now we have something with the units digit of 3. 3 times 7, 21. That’s a units digit of 1. So now we have a units digit of 1. 1 times 7, we have a units digit of 7.
7 times 7, 49. We have a units digit of 9. 9 times 7 we have 63. We have a units digit of 3. 3 times 7, 21. We have a units digit of 1.
And notice what a, what results here is a repeating pattern. It’s like mathematical wallpaper. We get 9, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, 1. It’s just gonna go on like that forever. It’s a repeating pattern. The period of the pattern, the period of the pattern, how many steps is it going to take to repeat.
Well it repeats in a batch of four. Every four it repeats. Now that’s very important, it’s very important to figure out the period of the pattern. Almost all these unit digit questions incidentally, the period will be four. If it happens that we’re doing powers of 9, that has a period of two, that makes things a little bit simpler.
But if the period of four, this is most typical for most of the unit digit questions you’ll see on the test, the period will be four. What that means is, every time we get to a multiple of four, when the power’s a multiple of four, we always come back to the same place. So 7 to the 4th has a unit digit of 1. 7 to the 8th has a unit digit of 1.
7 to the 12th, the 16th, the 20th, the 40th, the 80th, all of those would have unit digits of 1. That tells us how to extend the pattern. Because 120 is a multiple of 4. So that means 120 will have a units digit of 1, just like all the other multiples of 4.
And then we just continue the pattern from there. After 1, we have a units digit of 7. After 7, we have a units digit of 9. After 9, we have a units digit of 3. So that means 7 to the power of 123 has a units digit of 3. And therefore, any number ending in 7 to the power of 123 would have a units digit of 3.
So that means that our friend here, 57 to the power of 123, has a units digit of 3. And that’s the answer. Notice that we had to do absolutely nothing beyond single digit multiplication to answer this question. This question is always designed as a non-calculator mental math question. That’s very important to appreciate.
So the strategy for the unit digit question. Focus only on single-digit multiplication. Look for the repeating pattern and determine the period, and again, the most often the period is four, and then extend the pattern using multiples of the period. This is the technique we used to solve these questions.
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