$1-$ Let $a,b,$ and $c$ be integers such that $a0.$ Show that if $a|b|$ and $a|c|,$ then $a|(mb+nc)|$ whenever $m$ and $n$ are integers.

$2-$ Show that if $a-=b(modm)$ and $c-=d(modm),$ where $a,b,c,d$ are integers and $m$ is a positive integer, then $ac-=bd(modm).$

$3-$ Convert a number in a number system to another one. Write down the intermediate steps.

a$)$ Decimal number to Hexadecimal number:

$169150$

b$)$ Binary number to decimal number:

$(1011010\mathrm{.}101{)}_2$

c$)$ Hexadecimal number to octal number:

$(D8E3\mathrm{.}5A{)}_{16}$

$4-$ Calculate $A^{**}{B}^{**}C$ if it is defined. Write down the major intermediate steps.

$A=\left[\begin{array}{cccc}2& 5& 0& 1\\ 3& 1& 4& 3\\ 5& 2& 1& 4\end{array}\right],B=\left[\begin{array}{cc}3& 1\\ 6& 0\\ 2& 4\\ 1& 5\end{array}\right],C=\left[\begin{array}{ccc}1& -2& 3\\ 2& -1& 3\end{array}\right]$