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Prove that 8^n - 3^n is divisiible by 5 for all natural numbers n.
\n" ); document.write( "Do i use Induction?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "YES. You are right. You do use induction.\r
\n" ); document.write( "\n" ); document.write( "Let P(n) = 8^n - 3^n ----(I)
\n" ); document.write( "Consider n=1
\n" ); document.write( "Then 8^n - 3^n= 8^1-3^1 = 8-3 =5 which is divisible by.5----(1)
\n" ); document.write( "Consider n=2
\n" ); document.write( "Then 8^n - 3^n= 8^2-3^2 = 64-9 =55 =(11X5) which is divisible by.5
\n" ); document.write( "Let now P(m) =8^m-3^m ----(II)
\n" ); document.write( "be true for some intermediate positive integer m>2
\n" ); document.write( "Then P(m+1)= 8^m-3^m= (5+3)^m - 3^m
\n" ); document.write( "= {5^m +m(5^(m-1))X3 +[m(m-1)/2][(5^(m-2))X3^2]+......[m(5^1)(3^(m-1))]
\n" ); document.write( "+3^m}-3^m
\n" ); document.write( "= 5^m +m(5^(m-1))X3 +[m(m-1)/2][(5^(m-2))X3^2]+......[m(5^1)(3^(m-1))]
\n" ); document.write( "=5X[....]
\n" ); document.write( "{as (+3^m-3^m)=0 and each of the other terms contains 5 as a factor}
\n" ); document.write( "which is divisible by 5
\n" ); document.write( "Thus P(m+1) is true
\n" ); document.write( "Therefore P(m) implies P(m+1) ----(*)
\n" ); document.write( "And since P(2) is true therefore by (*),P(2+1)=P(3) is true
\n" ); document.write( "And since P(3) is true therefore by (*),P(3+1)=P(4) is true
\n" ); document.write( "And since P(4) is true therefore by (*),P(4+1)=P(5) is true
\n" ); document.write( "contiuning similarly finally we arrive at the proposition TRUE for any given positive integer n however large! \n" ); document.write( "