# Number System: Concept of Unit Digit

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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
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
• 2716987
• 15625369
• 190654789321
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

###### 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.