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
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.
Must Read Number System Articles
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.
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.
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.
_10.jpg?null)
_3.jpg?null)
-icon.jpg?null&itok=2FjKP8-O "Chain Rule Practice Problems"){kind=icon}
