unit digit

4¹ = 4

4² = 16

4³ = 64

Therefore 4²ⁿ = XXXXXXXX6     4²ⁿ⁺¹ = XXXXXXXX4

Therefore unit digit of 2354¹⁰⁴⁸ = unit digit of 4¹⁰⁴⁸  = 6.

unit digit of 2354 raised to the power of 1048???

\(\begin{array}{|rcll|} \hline && 2354^{1048} \pmod {10} \qquad & | \qquad 2354 \pmod{10} \equiv 4 \\ &\equiv& 4^{1048} \pmod {10} \\ &\equiv& 4^{2\cdot 524} \pmod {10} \\ &\equiv& (4^2)^{524} \pmod {10} \qquad & | \qquad 4^2 \pmod{10} \equiv 16 \pmod{10} \equiv 6 \\ &\equiv& 6^{524} \pmod {10} \qquad & | \qquad 6^n \pmod{10} \equiv 6 \\ &\equiv& 6 \pmod {10}\\ \hline \end{array} \)

Unit digit of 2354 raised to the power of 1048 is 6