01. A) Let a, b, and c be integers, where a = 0. Then (i) If a | b and a | c, then a | (b + c); (ii) If a | b, then a | bc for all integers c; (iii) If a | b and b | c, then a | c. Course Code: CSE-1102 B) Use Algorithm of Modular Exponentiation to find 1231001 mod 101

A.

(i) $a|b\Rightarrow b=ak_1. a|c\Rightarrow c=ak_2\Rightarrow (b+c)=a(k_1+k_2)\Rightarrow a|(b+c).$

(ii) $a|b\Rightarrow b=ak\Rightarrow bc=akc\Rightarrow a|bc.$

(iii) $a|b\Rightarrow b=ax. \ b|c\Rightarrow c=by=axy\Rightarrow a|c.$

B.

1231001= 123$\times 10^2\times 10^2 +1001= (22)\times (-1)\times (-1)+(101\times 10 -9)$ modulo 101

= 22-9=13.