Prove that 2 n + 3 n is a multiple of 5 for all odd n ∈ N

Question

Prove that       2          n        +      3          n       is a multiple of 5 for all odd   n  ∈  N

kilinumad · Accepted Answer

Prove that       2    n    +      3    n   is a multiple of 5 for all odd n   P  (  n  )  :      2    n    +      3    n    P  (  1  )  :      2    1    +      3    1    =  5 divisible by 5let   P  (  n  )  :      2    n    +      3    n   divisible by 5: that is       2    n    +      3    n    =  5.k for k in Nwe have to show that   P  (  n  +  1  )  :      2          n      +      1        +      3          n      +      1       divisible by 5  P  (  n  +  1  )  :      2          n      +      1        +      3          n      +      1        =      2          n        2  +      3          n        3  =      2    n    2  +      3    n    (  2  +  1  )  =      2    n    2  +  2      3    n    +      3    n    =  2  (      2    n    +      3    n    )  +      3    n    =  2.  5  k  +      3    n

moiraudjpdn · Accepted Answer

2          2      k      +      1        +      3          2      k      +      1        ∀  k  ∈  NNote that all odd numbers can be written as 4k+1 and 4k+3.Observe      2          4      k      +      1        ≡  2  mod  5 and      3          4      k      +      1        ≡  3  mod  5Also      3          4      k      +      3        ≡  2  mod  5 and       2          4      k      +      3        ≡  3  mod  5Now   2  mod  5  +  3  mod  5  =  5  mod  5  =  0  mod  5

Prove that 2^n +3^n is a multiple of 5 for all odd n in N

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