I2.1 If $x_{0}, y_{0}, z_{0}$ is a solution to the simultaneous equations below, $\left\{\begin{array}{l}x-y-z=-1 \\ y-x-z=-2 \\ z-x-y=-4\end{array}\right.$
I2.2 If $\beta$ is the reminder of $\underbrace{111 \cdots 111}_{10011 s} \div \alpha$, determine the value of $\beta$.
I2.3 If $\gamma$ is the remainder of $\left[(\beta-2)^{100}+\beta^{50}+(\beta+2)^{25}\right] \div 3$, determine the value of $\gamma$.
I2.4 If the equation $x^{4}+a x^{2}+b x+\delta=0$ has four real roots with three of them being $1, \gamma$ and $\gamma^{2}$, determine the value of $\delta$.
determine the value of $\alpha=x_{0}+y_{0}+z_{0}$.
$$
\begin{array}{l}
=7 \mathrm{~m}+7 \times 158+5 \text {, where } \mathrm{m} \text { is an integer } \\
\beta=5 \\
3^{100}+5^{50}+7^{25}=3^{100}+(6-1)^{50}+(6+1)^{25} \\
=3^{100}+6^{n}+1+6^{m}+1 \text {, where } m \text { and } n \text { are integers } \\
\gamma=2 \\
\end{array}
$$

Let the fourth root be $t$.
$$
\begin{array}{l}
1+2+2^{2}+t=\text { sum of roots }=-\frac{\text { coefficient of } x^{3}}{\text { coefficient of } x^{4}}=0 \\
t=-7 \\
1 \times 2 \times 2^{2} \times(-7)=\text { product of roots }=\frac{\text { constant term }}{\text { coefficient of } x^{4}}=\delta \\
\delta=-56
\end{array}
$$