Question

Show that 3.4^n+51 is divisible by 3 and 9 for all positive integers n.

Answer 1

Answer:

(3*4^n+51)

3(4^n+17)

Rem→3(4^n+17)/3 = 0

now checking for 9

(3*4^n+51)

3(4^n+17)/9 = (4^n+17)/3

(2^{2n}+18–1)/3

=Rem→(2^{2n}-1)/3+18/3 = (2^{2n}-1)/3+0

(2^2n-1)

3 = 4–1 = (2^{2*1}–1)

3 = 16-13 = (2^{2*2}-1–12)

3 = 64-61 = (2^(2*3)-1–60)

Hence, 3n = 4^{n}-1 = 2^{2n}-1 is divisible by 3 & 9 both.