(Canadian Senior Mathematics Contest 2024, Part B, Question 1, CEMC - UWaterloo)

(a) Determine the integer $x$ for which $2^{x+1} = 64$.
(b) Determine the values of $t$ and $u$ that satisfy the following system of equations:
$2^{t+u} = 2^8$
$5^{u-3} = 5^2$
(c) Determine the integers $m$ and $r$ for which $2^{2m+r}5^{3m-r} = 2^7 5^3$.
(d) This problem has been adapted by the MCR to fit the online experience.
Show that there are no integers $p$ and $q$ for which $2^p + 5^q = 80$. (do not answer)

Note: Submit your answers comma-separated for each part which needs to be answered. For part b, your answer should be the pair $(t,u)$, and for part c, your answer should be the pair $(m,r)$.