(Solved): please explain step by step.... how they get the negative number ( -5) and so on Example. Find \( 1 ...

please explain step by step.... how they get the negative number ( -5) and so on

Example. Find \( 17^{2022} \) modulo 98. Solution 1. We have \[ \begin{array}{c} 17^{2}=289 \equiv 93 \equiv-5 \quad(\bmod 98), \quad 17^{4} \equiv(-5)^{2}=25 \quad(\bmod 98) \\ 17^{8} \equiv 25^{2}=625 \equiv 37 \quad(\bmod 98), \quad 17^{16} \equiv 37^{2} \equiv-3 \quad(\bmod 98) \\ 17^{32} \equiv 9 \quad(\bmod 98), \quad 17^{64} \equiv-17 \quad(\bmod 98), 17^{128} \equiv-5 \quad(\bmod 98) \\ 17^{256} \equiv 25 \quad(\bmod 98), \quad 17^{512} \equiv 37 \quad(\bmod 98), \quad 17^{1024} \equiv-3 \quad(\bmod 98) \end{array} \] Hence, \[ \begin{aligned} 17^{2022} &=17^{1024+512+256+128+64+32+4+2} \\ & \equiv(-3)(37)(25)(-5)(-17)(9)(25)(-5) \equiv 71 \quad(\bmod 98) . \end{aligned} \] Solution 2. We start as before, but stop at \( 17^{128} \). Since \[ 17^{128} \equiv-5 \equiv 17^{2} \quad(\bmod 98) \quad \Longrightarrow \quad 17^{126} \equiv 1 \quad(\bmod 98), \] we get \[ 17^{2022}=17^{126 \cdot 16+6}=\left(17^{126}\right)^{16} 17^{4+2} \equiv 1^{16}(25)(-5) \equiv 71 \quad(\bmod 98) \]