Prove by induction that any natural number is either even or odd…. 1 answer below »

(1) Prove by induction that any natural number is either even or odd.

(2) Prove by induction that 32n+1 + 2n+2 is divisible by 7 for all n ? N.

(3) Prove by induction that every natural number n = 12 can be written

as a sum of 4s and 5s.

Hint: It is useful to note that if a number greater than 12 is written

as a sum of 4s and 5s, there must be at least three terms in that

sum.

(4) Fix a natural number n. Use induction to prove that (1 + n)

3m – 1

is divisible by n for every m ? N.

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