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Unit Digit

The application of the Number System extends to nearly every mathematical topic, underscoring its crucial significance. Within the Number System, various sub-topics 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:

  • 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.
Important Concepts: Number System - 3 - CAT

Number of Zeroes

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:

  • 2n * 5m gives n number of zeros if m>n
  • 2n * 5m gives m number of zeros if n>m

Shortcut Methods

Make sure you follow these given shortcut methods for various topics to excel in the exam:

  • Sum of first n natural numbers = [n(n+1)] / 2
  • Sum of squares of first n natural numbers = [n(n+1)(2n+1)] / 6
  • Sum of cubes of first n natural numbers = [n(n+1)]2 / 4
  • Sum of first n even natural numbers = n(n+1)
  • Sum of first n odd natural numbers = n2
  • Sum of squares of first n even natural numbers = [2n(n+1)(2n+1)] / 3
  • Sum of squares of first n odd natural numbers = [2n(2n+1)(2n-1)] / 3
  • The calculation of digital sum is very easy to understand. It simply refers to the process of repeated addition of the digits of the number until a single digit is left. For example, the digital sum of 34678 will be 3 + 4+ 6 + 7 + 8 = 28 = 2 + 8 = 10 = 1 + 0 = 1. Hence, 1 is the digital root of 34678.
  • When (x-1)n is divided by x, then the remainder is:

1, if n is an even natural number
x-1, if n is an odd natural number.

  • The shortcut trick for finding the unit digit of a given number of the form Xn is to divide power ‘n’ by the cyclicity of the number X and find the unit digit for the remaining power of the number X.

For e.g. for finding the unit digit of 235
Divide 35 by 4(cyclic power of 2)
The remaining number will be 3
So, the unit digit will 23 = 8

  • When the digits of a 3- digit number are reversed; then the difference between the given number and the reversed number is always divisible by 99.
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