Units Digit Questions
So for example, a typical question of this sort, what is the unit 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 were really fascinated with that sort of thing. But the point is no calculator's gonna help you with this. Now, at this point, this probably really blows your mind, why is the test asking you to do something that no calculator on the planet can do? Well let's be careful here. No calculator on the planet could figure out that entire power, all 216 digits, but we don't need 216 digits.
We only need the unit's 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 going to 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 will need a 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 time some time to sink in.
Let's think about this for a second. 3 times 6, you know, is 18. So, we multiply a unit digit of a 3 times the unit digit of 6. I get a units 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 the product. Now this is a hard idea to understand. What I strongly suggest is stop this video, pick up a calculator and actually try this for yourself.
Enter some numbers, say enter a number in the hundreds that has a unit digit of 3. Enter another number in the hundreds that has a unit digit of 6. Multiply them together. Time and time again you will get some bigger number with a units digit of 8. You'll always get that same units digit answer. Then try some other combinations of digits.
Unit digit of 2, unit digit of 7. Or unit digit of 3, unit digit of 9. Something like that, just pick your own combination. Do the single digit multiplication. So, 3 times 9 for example's 27. So unit digit of 3 times the unit digit of 9 will always have a unit digit of 7.
Again experiment with this and verify for yourself on your calculator that it works. If you're not a mathy sorta person. One way that you build intution or 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 granddad question here, 57 of 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 going to 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 multiplied by 7 again, 9 times 7, 63. So now, we have something with a 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. 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 does it take to repeat. Well it repeats in a batch of four. Every 4, 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 4. If it happens that we're doing powers of 9, that has a period of 2. That makes things a little bit simpler. But if it's a period of 4, this is most typical for most of the unit. Digit question 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 powers are a multiple of four, we always come back to the same place. So seven to the fourth has a unit digit of one. Seven to the eighth has a unit digit of one. Seven to the 12th, the 16th, the 20th, the 40th, the 80th, all of those would have unit digits of one.
That tells us how to extend the pattern. Because 100 and 20 is a multiple of four. So that means 100 and 20 will have a units digit of one just like all the other multiples of four. And then we just continue the pattern from there. After one we have a units digit of seven.
After seven we have a unit digit of nine. After nine 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 units digit question.
Focus only on single digit multiplication. Look for the repeating pattern to determine the period, and again most often the period is four, and then extend the pattern using multiples of the period. This is the technique we use to solve these questions.
FAQ: What is a “units digit”? Is it the same as the “ones’ place”?
A: Yes! The "units digit" is the same as the "ones’ place”. A units digit is simply the right-most digit in any number (but to the left of the decimal point) --- for example, the number 28.435 has a units digit of 8.
FAQ: When we’re asked to find the units digit of 57^123, how do we know to focus on 7^120 because 120 is a multiple of 4? What does that have to do with anything?
A: The key is that the pattern repeats after four periods. That's why we look for a multiple of four, not six or three, or some other number.
We worked out the units digit pattern to:
7^1 = 7
7^2 = *9
7^3 = *3
7^4 = *1
7^5 = *7...the pattern has restarted
This is a pattern of 7, 9, 3, 1. So, it's important to note that there are four numbers in the pattern.
Because there are four periods before the pattern repeats, to find 7^123, we want to find what's left over when we divide 123 by 4. We know that every power of 4 will have a units digit of 1 because of the pattern we found earlier. Then we can just count up or down the pattern from there.
Another example: Let's say I had some repeating pattern, with 6 things in the pattern:
1st: 4
2nd: 5
3rd: 8
4th: 3
5th: 1
6th: 9
I want to find the 602nd number in the pattern. Well, NOW I would look for a multiple of 6, because there are six periods in the repeating pattern.
So I find a multiple of 6 just before 602...600. I know that number is 9. Now I just count up two ...4....5 and I know the 602nd number is 5.
To summarize the general strategy for solving this kind of problem:
1) Find the number of periods "n" in the pattern
2) Note what the value is for the last term in the repeating pattern
3) Finding a power that is divisible by "n" will have that same value as the last term in the repeating pattern
4) So if you're trying to find some large power, find the closest multiple of "n" to that power and count through the pattern from there.
FAQ: Could the GRE also ask us to find the tens digit of a number raised to another power?
A: You might be asked to do this, but it’s very unlikely! These kinds of questions will be almost exclusively units-digits questions. On a rare occasion you might be asked about a tens-digit, but you should be able to deduce this fairly reasonably. For example, imagine the following question:
What's the tens-digit of 5^35?
Let’s look at the pattern here, from 5^2
5^2 = 25 5^3 = 125 5^4 = 625 5^5 = 3125
See the pattern emerge? It's always 2! So we know that 5^35 has a tens digit of 2.
Keep in mind that the GRE is just testing your ability to pattern match, so if you're given something like this, you should look for the pattern.