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Finding the Unit Digits of Large Powers | Quantitative Reasoning for GRE PDF Download

Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you’re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on difficult units digits.)
See how you do with this question:

Q. What is the units digit of 5745?
A) 1
B) 3
C) 5
D) 7
E) 9

Ans: To solve this, we’ll begin examining smaller powers and look for a pattern.
571 = 57 (the units digit is 7)
572 = 3,249 (the units digit is 9)
573 = 185,193 (the units digit is 3)
Aside: Since these powers increase quickly, it’s useful to notice that we need only multiply the units digit each time. For example, the units digit of 572 is the same as the units digit of 72. Similarly, the units digit of 573 is the same as the units digit of 73.
So, once we know that the units of 572 is 9, we can find the units digit of 573 by multiplying 9 by 7 to get 63. So the units digit of 573 is 3.
To find the units digit of 574, we’ll multiply 3 by 7 to get 21. So the units digit of 574 is 1.
When we start listing the various powers, we can see a pattern emerge:
The units digit of 571 is 7
The units digit of 572 is 9
The units digit of 573 is 3
The units digit of 574 is 1
The units digit of 575 is 7
At this point, we should recognize that the pattern begins to repeat. The pattern goes: 7-9-3-1-7-9-3-1-7-9-3-1-…
Since the pattern repeats itself every 4 powers, we say that the “cycle” equals 4
Now comes an important observation:

  • The units digit of 571 is 7
  • The units digit of 572 is 9
  • The units digit of 573 is 3
  • The units digit of 574 is 1
  • The units digit of 575 is 7
  • The units digit of 576 is 9
  • The units digit of 577 is 3
  • The units digit of 578 is 1
  • The units digit of 579 is 7
  • The units digit of 5710 is 9
  • The units digit of 5711 is 3
  • The units digit of 5712 is 1. . . etc.

As you can see, since the cycle = 4, the units digit of 57k will be 1 whenever k is a multiple of 4.
Now to find the units digit of 5745, all we need to do is recognize that the units digit of 5744 is 1 (since 44 is a multiple of 4).
From here, we’ll just continue with our pattern:
The units digit of 5744 is 1
The units digit of 5745 is 7
The units digit of 5746 is 9
The units digit of 5747 is 3 . . . etc.
So, the units digit of 5745 is 7, which means the answer is D.

 

Practice Question:

Q. What is the units digit of 8375?
Ans: 7

Q. What is the units digit of 3961?
Ans: 9

The document Finding the Unit Digits of Large Powers | Quantitative Reasoning for GRE is a part of the GRE Course Quantitative Reasoning for GRE.
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FAQs on Finding the Unit Digits of Large Powers - Quantitative Reasoning for GRE

1. How do you find the unit digit of a large power?
Ans. To find the unit digit of a large power, you need to look for a pattern in the unit digits of the powers of the base number. For example, if the base number ends in 2, the unit digit of any power of that number will alternate between 2 and 8. By identifying and analyzing such patterns, you can determine the unit digit of a large power.
2. What is the significance of the unit digit in mathematics?
Ans. The unit digit in mathematics is significant as it helps determine the cyclicity or repeating pattern of numbers. It is especially useful when dealing with large powers, as the unit digit often repeats after a certain number of iterations. By understanding the unit digit pattern, we can simplify calculations and solve problems more efficiently.
3. Can the unit digit of a large power be zero?
Ans. Yes, the unit digit of a large power can be zero in certain cases. For example, any power of 10 (such as 10^2, 10^3, 10^4, etc.) will have a unit digit of zero. Similarly, powers of any number ending in zero will also have a unit digit of zero. However, it is important to note that not all large powers will have a unit digit of zero.
4. Are there any exceptions to finding the unit digit of large powers?
Ans. Yes, there can be exceptions to finding the unit digit of large powers. Some numbers may not follow a specific pattern or may have irregularities in their unit digit sequence. In such cases, it becomes more challenging to determine the unit digit without additional information or techniques. It is always recommended to thoroughly analyze the number and its powers to identify any exceptions.
5. What strategies can be used to find the unit digit of large powers?
Ans. There are several strategies that can be used to find the unit digit of large powers. These include identifying patterns, using modular arithmetic, applying cyclicity rules, or utilizing properties of specific numbers. For example, the unit digit of powers of 2 follows a pattern of alternating between 2 and 8. By understanding and applying these strategies, you can efficiently determine the unit digit of large powers.
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