Prove by induction that 64n 6 mod 9 For a base case Im assum

Prove by induction that 6(4ⁿ) 6 (mod 9)

For a base case, I\'m assuming you left n=1 and show that 24 6(mod 9) ? But I\'m not sure about that. And then I have idea what to do for the inductive step. Not sure if it\'s strong induction or not. Help please!

Solution

Base Case: (n=1)

6(4(1)) = 24 mod(9) = 6

Hence the base case is satisfied

Induction Step (n=k)

6(4k) = 24k

assuming that the base case is true, then we will have 24kmod(9) = 6

Hypothesis Step (n=k+1)

6(4(k+1) = 6(4k+4) = 24k + 24

(24k+24)mod(9) = (24kmod9 + 24mod9) = 6

Hence the hypothesis step is proved, therefore the induction is satisfied