Question

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

Answer 1

Answer:

$32n\,\,mod\,\,13=6n;\,\,n=0,1,2,3,...$

Step-by-step explanation:

$a\,\,mod\,\,b$ refers to the quotient that is obtained on dividing $a$ by $b$.

To find: $32n\,\,mod\,\,13;\,\,n=0,1,2$

Solution:

For n = 0:

$32n\,\,mod\,\,13=0\,\,mod\,\,13=0$

For n = 1:

$32n\,\,mod\,\,13=32\,\,mod\,\,13=6$

For n = 2:

$32n\,\,mod\,\,13=64\,\,mod\,\,13=12$

Therefore,

$32n\,\,mod\,\,13=0\,\,mod\,\,13=0=0\times 6\\32n\,\,mod\,\,13=32\,\,mod\,\,13=6=1\times 6\\32n\,\,mod\,\,13=64\,\,mod\,\,13=12=2\times 6$

To find: general formula for $32n\,\,mod\,\,13$

So, as per the pattern, $32n\,\,mod\,\,13=6n;\,\,n=0,1,2,3,...$