Will calculator help?

$\begin{aligned} 5^{44}+6 & \equiv \left(5^{\color{#3D99F6} 44 \text{ mod }\phi (8)} + 6 \right) \text{ (mod 8)} & \small \color{#3D99F6} \text{Since }\gcd(5,8) = 1 \text{, Euler's theorem applies.} \\ & \equiv \left(5^{\color{#3D99F6} 44 \text{ mod }4} + 6 \right) \text{ (mod 8)} & \small \color{#3D99F6} \text{Euler's totient function }\phi (8) = 4 \\ & \equiv \left(5^{\color{#3D99F6}0} + 6 \right) \text{ (mod 8)} \\ & \equiv 7 \text{ (mod 8)} \end{aligned}$

$\implies 5^{44}+6 + \boxed{1} \equiv 0 \text{ (mod 8)}$ .

$5^{44}+6 \equiv \left(-3\right)^{44}+6\equiv3^{44}+6\equiv9^{22}+6\equiv1^{22}+6\equiv1+6\equiv7\equiv-1\pmod{8}\implies8\mid5^{44}+6+1$ .

Hence, the answer is $\boxed{1}$ .

Note that $\large { 5 }^{ 2 }\equiv 1\quad mod\quad 8$

The Congruence can be written as $\large 1+6\equiv \quad mod\quad 8$

$\large \therefore \text{the sum is equivalent to} 7 \hspace{2cm} \text{Hence 1 should be added}$

5^(even)gives you last digits as 625 where as 5^(odd) gives you 125.. as 5^44 is even power 625 are the last 3 digits which are needed to check if number is divisble by 8 .remainder is 7 so add 1.