CYCLICITY

CYCLICITY
Digits 4 & 9 : Both these numbers have a cyclicity of only two different digits as their unit’s digit.
    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.
The number in the form of 4ⁿ,\(n\in z\)

Example : i) The unit’s place digit in the expansion of 4¹⁹⁶⁷ is 4.
  ii) The unit’s place digit in the expansion of 4²⁰⁰⁸ is 6. 
    Likewise, the powers of 9 operate as follows:
    9¹=9
   9²=18
   9³=27 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
        \({194^{65478932}}\)
        Answer = 6 (since power is even)
The number in the form of 9ⁿ,\(n\in z\)
        
Example : i) The unit’s place digit in the expansion of 9¹⁰⁰ is 1
        ii) The unit’s place digit in the expansion of 9²⁰⁰⁷ is 9.