Question

g A. (Points: 7) Compute (without using a calculator) 241^257 mod 12 B. (Points: 3) Compute Z*20 C. (Points: 6) Find the multiplicative inverse of 7 in Z19

Answer 1

Answer:

$241^{257}\ mod\ 12 =1$

$7 * 20 = 140$

$\frac{1}{700}$

Step-by-step explanation:

Solving (a): 241^257 mod 12

To do this, we simply calculate $241\ mod\ 12$

Because $a\ mod\ b = a^n\ mod\ b$

The highest number less than or equal to 241 that is divisible by 12 is 240; So:

$241\ mod\ 12 = 241- 240$

$241\ mod\ 12 =1$

Hence:

$241^{257}\ mod\ 12 =1$

Solving (b): 7 * 20

$7 * 20 = 140$

Solving (c): Multiplicative inverse of 7 in 719

The position of 7 in 719 is 700

So, the required inverse is 1/700 ---- i.e. we simply divide 1 by the number