exponent

what are the last two digits of 4^2000 ?

or \(4^{2000} \equiv x \pmod {100}\)

\(\boxed{~ 4^{11} \equiv 4 \pmod{100} ~} \)

\(\small{ \begin{array}{rcl} && 4^{2000} \pmod {100} \\ &\equiv& 4^{11\cdot 181 + 9} \pmod {100}\\ &\equiv& 4^{11\cdot 181} \cdot 4^9 \pmod {100}\\ &\equiv& (\underbrace{4^{11}}_{\equiv 4 \pmod {100}})^{181} \cdot 4^9 \pmod {100}\\ &\equiv& 4^{181}\cdot 4^9 \pmod {100}\\\\ &\equiv& 4^{11\cdot 16 + 5} \pmod {100}\\ &\equiv& 4^{11\cdot 16} \cdot 4^5 \cdot 4^9 \pmod {100}\\ &\equiv& (\underbrace{4^{11}}_{\equiv 4 \pmod {100}})^{16} \cdot 4^5 \cdot 4^9 \pmod {100}\\ &\equiv& 4^{16} \cdot 4^5 \cdot 4^9 \pmod {100}\\ &\equiv& 4^{30} \pmod {100}\\\\ &\equiv& 4^{11\cdot 2 + 8} \pmod {100}\\ &\equiv& 4^{11\cdot 2} \cdot 4^8 \pmod {100}\\ &\equiv& (\underbrace{4^{11}}_{\equiv 4 \pmod {100}})^{2} \cdot 4^8 \pmod {100}\\ &\equiv& 4^{2} \cdot 4^8 \pmod {100}\\ &\equiv& 4^{10} \pmod {100}\\ &\equiv& 76\pmod {100} \end{array} }\)

The last two digits of \(4^{2000}\) is 76

heureka: Isn't 4 raised to ANY power ending in "0", always going to end in "76"?, i.e, 4^10, 4^70, 4^160, 4^500, 4^1000 and 4^2000......etc. But why?