Finding the Unit Digit of a^b

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In summary, to find the unit digit of a^b, you can use the concept of cyclicity to identify a repeating pattern of unit digits. The cyclicity of a base number refers to the number of unique unit digits that appear when the number is raised to different powers. The unit digit of a^b can be zero if the base number's unit digit is zero and the power b is greater than 1. If the power b is negative, the expression can be rewritten as 1/(a^b) and the unit digit can be found using the same method. This concept is useful in various real-world applications such as cryptography, computer programming, and number theory.
  • #1
PrudensOptimus
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Hi, is there any way to find Unit Digit of a expression, say

[tex]a^b[/tex]


where a, b, are positive integers?
 
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  • #2
Yes.

Hint: fix a, vary b, and ignore the irrelevant details, see if you can find a clue...
 
  • #3
U mean a pattern?

Like 2^2 = 4, like 2^2222 = 4? Because 2222 is divisible by 2?
 
  • #4
I mean something a little more rigorous. :smile:

Consider the values of 2^1, 2^2, 2^3, ...
 

1. How do you find the unit digit of a^b?

To find the unit digit of a^b, you can use the concept of cyclicity. This means that every number has a repeating pattern of unit digits when raised to a power. For example, the unit digit of 2^1 is 2, 2^2 is 4, 2^3 is 8, and then the pattern repeats. By identifying the pattern for the base number, you can determine the unit digit of a^b by finding the remainder when b is divided by the cyclicity of the base number.

2. What is the cyclicity of a base number?

The cyclicity of a base number refers to the number of unique unit digits that appear when the number is raised to different powers. For example, the cyclicity of 2 is 4, as there are 4 unique unit digits (2, 4, 8, and 6) that appear when 2 is raised to different powers. The cyclicity of a base number can be determined by looking at the repeating pattern of unit digits when the number is raised to different powers.

3. Can the unit digit of a^b be zero?

Yes, the unit digit of a^b can be zero. This occurs when the unit digit of the base number is 0 and the power b is greater than 1. For example, the unit digit of 10^2 is 0, as the unit digit of 10 is 0 and 2 is greater than 1.

4. What happens if the power b is negative?

If the power b is negative, you can use the property of exponents to rewrite the expression as 1/(a^b). Then, you can find the unit digit of 1/(a^b) using the same method as finding the unit digit of a^b. However, you must remember to take the reciprocal of the unit digit found for a^b.

5. How can finding the unit digit of a^b be useful in real-world applications?

Finding the unit digit of a^b can be useful in a variety of real-world applications. For example, it can be used in cryptography to encrypt and decrypt messages, in computer programming to optimize algorithms, and in number theory to solve complex mathematical problems. Additionally, understanding the cyclicity of different numbers can also help in identifying patterns and making predictions in various fields such as finance and economics.

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