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Question 29235: Prove that 8^n - 3^n is divisiible by 5 for all natural numbers n.
Do i use Induction?
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! Prove that 8^n - 3^n is divisiible by 5 for all natural numbers n.
Do i use Induction?
YES. You are right. You do use induction.
Let P(n) = 8^n - 3^n ----(I)
Consider n=1
Then 8^n - 3^n= 8^1-3^1 = 8-3 =5 which is divisible by.5----(1)
Consider n=2
Then 8^n - 3^n= 8^2-3^2 = 64-9 =55 =(11X5) which is divisible by.5
Let now P(m) =8^m-3^m ----(II)
be true for some intermediate positive integer m>2
Then P(m+1)= 8^m-3^m= (5+3)^m - 3^m
= {5^m +m(5^(m-1))X3 +[m(m-1)/2][(5^(m-2))X3^2]+......[m(5^1)(3^(m-1))]
+3^m}-3^m
= 5^m +m(5^(m-1))X3 +[m(m-1)/2][(5^(m-2))X3^2]+......[m(5^1)(3^(m-1))]
=5X[....]
{as (+3^m-3^m)=0 and each of the other terms contains 5 as a factor}
which is divisible by 5
Thus P(m+1) is true
Therefore P(m) implies P(m+1) ----(*)
And since P(2) is true therefore by (*),P(2+1)=P(3) is true
And since P(3) is true therefore by (*),P(3+1)=P(4) is true
And since P(4) is true therefore by (*),P(4+1)=P(5) is true
contiuning similarly finally we arrive at the proposition TRUE for any given positive integer n however large!
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