Number System: Concept of Unit Digit

Number System has its application in almost every other topic in mathematics. This very much defines the importance of this topic. Number System mainly includes further sub-topics like HCF and LCM, unit digit, factors, cyclicity, factorials, Euler number, digital root, etc.
To understand the concept of unit digit, we must know the concept of cyclicity . This concept is mainly about the unit digit of a number and its repetitive pattern on being divided by a certain number
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The concept of unit digit can be learned by figuring out the unit digits of all the single digit numbers from 0 - 9 when raised to certain powers.
These numbers can be broadly classified into three categories for this purpose:
1. Digits 0, 1, 5 & 6: When we observe the behaviour of these digits, they all have the same unit's digit as the number itself when raised to any power, i.e. 0^n = 0, 1^n =1, 5^n = 5, 6^n = 6. Let's apply this concept to the following example.
Example: Find the unit digit of following numbers:
  • 185563
    Answer= 5
  • 2716987
    Answer= 1
  • 15625369
    Answer= 6
  • 190654789321
    Answer= 0
2. Digits 4 & 9: Both these numbers have a cyclicity of only two different digits as their unit's digit.
Let us take a look at how the powers of 4 operate: 41 = 4,
42 = 16,
43 = 64, and so on.
Hence, the power cycle of 4 contains only 2 numbers 4 & 6, which appear in case of odd and even powers respectively. 
Likewise, the powers of 9 operate as follows:
91 = 9,
92 = 81,
93 = 729, and so on.
Hence, the power cycle of 9 also contains only 2 numbers 9 & 1, which appear in case of odd and even powers respectively.
So, broadly these can be remembered in even and odd only, i.e. 4odd = 4 and 4even = 6. Likewise, 9odd = 9 and 9even = 1.
Example: Find the unit digit of following numbers:
  • 189562589743
    Answer = 9 (since power is odd)
  • 279698745832
    Answer = 1(since power is even)
  • 154258741369
    Answer = 4 (since power is odd)
  • 19465478932
    Answer = 6 (since power is even)
3. Digits 2, 3, 7 & 8: These numbers have a power cycle of 4 different numbers.
21 = 2, 22 = 4, 23 = 8 & 24 = 16 and after that it starts repeating.
So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.
31 = 3, 32 = 9, 33 = 27 & 34 = 81 and after that it starts repeating.
So, the cyclicity of 3 has 4 different numbers 3, 9, 7, 1.
7 and 8 follow similar logic.
So these four digits i.e. 2, 3, 7 and 8 have a unit digit cyclicity of four steps.
Cyclicity Table
The concepts discussed above are summarized in the given table.
Number Cyclicity Power Cycle
1 1 1
2 4 2, 4, 8, 6
3 4 3, 9, 7, 1
4 2 4, 6
5 1 5
6 1 6
7 4 7, 9, 3, 1
8 4 8, 4, 2, 6
9 2 9, 1
10 1 0
Suggested Test :
Test-Check your understanding of the topic Number systems by taking this test now.
 
Solved Examples
Example 1: Find the Unit digit of 287562581
SolutionStep 1: We know that the cyclicity of 7 is 4.
Step 2: Divide the power 562581 by 4.
By doing that, we get a remainder=1.
Step 3: 1st power in the power cycle of 7 is 7.
Hence, the answer is 7.
Example 2: Find the Unit digit of 13445 * 54336
Solution: Cyclicity of 5 & 6 is 1. Since 5*6=30, the unit digit of given expression is 0.
Suggested Reading :
Learn the basics of Number System along with some shortcuts and tricks to handle the related questions.
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Key Learning:
  • You must remember the power cycles of all the digits from 1-10.
  • If the power cycle of number has 4 different digits, divide the power by 4, find the remaining power and calculate the unit’s digit using that. Similarly, if the power cycle of number has 2 different digits, divide the power by 2, find the remaining power and calculate the unit’s digit using that.
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