Brilliant Squared!

First, we will write the number $BRILLIANT$ from base 76 into base 10. We have

$\begin{aligned} B &= 11 \\ R &= 27 \\ I &= 18 \\ L &= 21 \\ A &= 10 \\ N &= 23 \\ T &= 29. \end{aligned}$

Thus,

$\begin{aligned} BRILLIANT_{76} &= (11 \times 76^8) + (27 \times 76^7) + (18 \times 76^6) + (21 \times 76^5) \\ & \quad + (21 \times 76^4) + (18 \times 76^3) + (10 \times 76^2) + (23 \times 76) \\ & \quad + 29. \end{aligned}$

The number 76 has the property that $76^n \equiv 76 \pmod{100}$ for all $n \geq 1$ . This can be seen by noting that $76^2 \equiv 76 \pmod{100}$ , then applying induction. Using this, we can write

$\begin{aligned} BRILLIANT_{76} &\equiv (11 \times 76) + (27 \times 76) + (18 \times 76) + (21 \times 76) \\ & \quad + (21 \times 76) + (18 \times 76) + (10 \times 76) + (23 \times 76) \\ & \quad + 29 \\ & \equiv 76(11 + 27 + 18 + 21 + 21 + 18 + 10 + 23) + 29 \\ & \equiv 76(149) + 29 \\ & \equiv 11353 \\ & \equiv 53 \! \! \! \! \pmod{100}. \end{aligned}$

Thus,

$(BRILLIANT_{76})^2 \equiv 53^2 \equiv 2809 \equiv \boxed{9} \! \! \! \! \pmod{100}$