The application of the Number System extends to nearly every mathematical topic, underscoring its crucial significance. Within the Number System, various subtopics such as HCF and LCM, unit digits, factors, cyclicity, factorials, Euler number, digital root, etc., are encompassed.
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:
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: 4^{1} = 4,
4^{2} = 16,
4^{3} = 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:
9^{1} = 9,
9^{2} = 81,
9^{3} = 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. 4^{odd} = 4 and 4^{even} = 6. Likewise, 9^{odd} = 9 and 9^{even} = 1.
Example: Find the unit digit of following numbers:
3. Digits 2, 3, 7 & 8: These numbers have a power cycle of 4 different numbers.
2^{1} = 2, 2^{2} = 4, 2^{3} = 8 & 2^{4} = 16 and after that it starts repeating.
So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.
3^{1} = 3, 3^{2} = 9, 3^{3} = 27 & 3^{4} = 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.
The concepts discussed above are summarized in the given table.
The number of zeros at the end of the product of any numbers is determined by the number of combinations of ‘2 * 5’ provided the below conditions are satisfied:
Make sure you follow these given shortcut methods for various topics to excel in the exam:
1, if n is an even natural number
x1, if n is an odd natural number.
For e.g. for finding the unit digit of 2^{35}
Divide 35 by 4(cyclic power of 2)
The remaining number will be 3
So, the unit digit will 2^{3} = 8
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