CYCLICITY AND CONGRUENCE MODULO

CYCLICITY AND CONGRUENCE MODULO
CYCLICITY
1. This concept is mainly about the unit digit of a number and its repetitive  pattern on being divided by a certain number
2. The concept of cyclicity can be learned by figuring  out  the unit digits of all the single digit numbers from 0-9 when raised to certain powers
3. Digits 0,1,5 and 6
      Observe the following pattern 

Also,
        6¹ = 6
        6² = 36
        6³ = 216
        .
        .
        .
        6ⁿ = -----6
When we observe the behaviour of these numbers, they all  have the same unit’s digits as the number itself when raised to any power
4. Example : Find the unit’s digits of the following numbers 
            185²⁰²² = 5
            166²⁰¹⁴ = 6
            156²⁰²⁰ = 6
            190³⁴²⁸¹ = 0
            271³⁴⁵  = 1
05. The unit’s digit of  the number 5n, where  is always 5. i.e., when we raise the power of 5 to any natural number , the resultant number contains 5 in its unit place

06. The unit’s digit of the number 6n, where \(n\in N\) is always 6 i.e., when we raise the power of 6 to any natural number, the resultant number contains 6 in its unit place.

07. Digit 4:
         Observe the following numbers
              4¹ = 4
              4² = 16
              4³ = 64 
              4⁴ = 256
              4⁵ = 1024
Here, clearly we observe if the power of 4 is odd number, then the unit’s digit is 4 and the power of 4 is even number, then the unit’s digit is 6.

8. Digit 9 : Observe the following pattern of number
        9¹ = 9
        9² = 18
        9³ = 729
        9⁴ = 6561
and so on.
    Here, clearly we oberve, if the power of 9 is odd number, the unit’s digit is 9 and the power of 9 is even number, then the unit’s digit is 1.

Example: (i)  The unit’s digit in the expansion of 9²⁰²² is 1. 
          (ii)   The unit’s digit in the expansion of 9²⁰²³ is 9
9. Similarly for the digit 2 we have 

Example (1) : The unit’s digit in the expansion of 2²⁰²² is 4. 
            Since 2022 = 4(505)+2 
                    = 4K + 2
Example (2) : The unit’s digit in the expansion of 2¹⁹⁷⁶ is 6.
            Since 1976 = 4 x 494
10. Digit 3 :

Example (1) : The unit’s digit in the expansion of 3¹⁹³⁴ is 9, 
            Since 1934 = 4(483) + 2
                    = 4k + 2
Example (ii) : The unit’s digit in the expansion of 3²⁰²³ is 7, 
            Since 2023 = 4(505) + 3
                    = 4K+
11. Digit 7 :

Example (i) : The unit’s digit in the expansion of 7¹³¹ is 3.
            Since 7¹³¹ = 74(32)+3
                    = 74(32) x 7
The unit’s digit place 7³ =3
12. Digit 8 :

Example 1: The unit’s digit in the expansion of 8²⁰⁴³ is 2.
            Since 2043 = 4(510)+3
                    = 4K+3
Example 2: The unit’s digit in the expansion of 8²⁰²² is 4.
Since 2022 = 4(505)+2     = 4K + 2
13. Power cycle of 0 \(\to\) 0
      Power cycle of 1 \(\to\) 1
      Power cycle of 2 \(\to\) 2, 4, 8, 6
      Power cycle of 3 \(\to\) 3, 9, 7, 1
      Power cycle of 4 \(\to\) 4, 6
      Power cycle of 5 \(\to\) 5
      Power cycle of 6 \(\to\) 6
      Power cycle of 7 \(\to\) 7, 9, 3, 1
      Power cycle of 8 \(\to\) 8, 4, 2, 6
      Power cycle of 9 \(\to\) 9, 1