prove that pn 1 is divisible by 8 for all integers n0Soluti

prove that p^n - 1 is divisible by 8, for all integers n>=0

Solution

9^n - 1 is divisible by 8

Let us prove this by induction

Consider n =1

Then 9-1 =8 is divisible by8.

True forn=1

Let it be true for n=k

i,e,9^k-1 is divisble by 8

Then 9^k-1 =8m forsome integer m

Or 9^k=8m+1

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Tp prove thatP(k+1) istrue

P(k+1) is to prove that 9^k+1 -1is divisible by 8

9^{k+1 =}9(9^k)

= 9(8m+1) =72m+9 =8(9m+1)+1

Thus we see that 9^k+1-1 = 8a forsome integer a

Thus proved for all integers n>=0