$1.(4$ points$)$ Find a simple function g of the smallest order so that f$($x$)$ is big$-$O of g$($x$).$ In your estimation you can use theorems learned in lecture about the big$-$O estimates of sums$/$products of functions.

f$($x$)=($x$2\mathrm{.}3+$ xlnx$5)(1\mathrm{.}1$x$+1+1\mathrm{.}2$x$)+($x$2+1\mathrm{.}2$x$)($x$3+0\mathrm{.}92$x$)$

$2.(4$ points$)$ Is there a smallest positive real number a for which x$26$x is big$-$O of ax$?$ Explain your answer.

$3.(1$ point$)$ Carry out the hexadecimal addition CA$10+4$F$57$ directly in hexadecimal. Do not convert the numbers to decimal, add them in decimal and convert the sum back to hexadecimal.

$4.(1$ point$)$ Carry out the binary multiplication $11011$ times $1001$ using only left bit shifts and addition.

$5.(4$ points$)$ Use the Euclidean Algorithm to demonstrate that $621$ and $82$ are relatively prime to each other. Explain.

$6.(4$ points$)$ Given an integer n$,$ show that you can multiply n by $35$ using only five multiplications by $2,$ two additions and storing intermediate results in memory.

$7.(2$ points$)$ Explain how you multiply an octal number n directly in octal by $(10)8.$ Illustrate your general explanation with n $=(741)8.$

$8.(2$ points$)$ Explain how you divide an octal number directly in octal by $(10)8.$ Here, divide means to carry out the division algorithm and to find both quotient and remainder. Illustrate your general explanation with n $=(741)8.$

$9.(4$ points$)$ Carry out the fast exponentiation algorithm to evaluate $32$n mod $13$ for n $=0,1$ and $2.$ These three values suffice to derive a formula for $32$n mod $13$ for all non$-$negative integers n$.$ State this formula and explain.

$10.(4$ points$)$ Give a closed$-$form formula in term of n for the value of the hex number in terms of n$.$ Your answer must not contain unresolved

n digits sigma sums. Show your work.

$11.($extra credit, $6$ points$)$ Write a Python program that systematically finds and prints all integers a$,$ from a $=1$ to $100,$ for which the function fa : $\{1,2,..,100\}->\{1,2,.\mathrm{.}100\}$;f$($x$)=$ ax mod $101$ is bijective. Show your program and the output of your program.