An integer n is called k-perfect if u(n) =kn. Note that a perfect number is 2-perfect. Show that 120 = 23 • 3 · 5 is 3-perfect.

You are given the following definitions to help you: O(g(n)) = {f(n): there exist positive constants c and no such that 0 ≤ f(n) ≤ cg(n) for all n ≥ no} . N(g(n)) = {f(n): there exist positive constants c and no such that 0 ≤ cg (n) ≤ f(n) for all n ≥ no} . ©(g(n)) = {f(n): there exist positive constants C₁, C2, and no such that 0 ≤ c₁g(n) ≤ f(n) ≤ c₂g (n) for all n ≥ no} . Using above definitions, prove that: 5. T(n) = 6n + 4n+ 3 € 0 (n) 6. T(n) = 100n + 10000 € O(n²) 7. T(n) = 5n² - 2n + 16 € O(n³) 8. T(n)=5n² - 2n + 16 is not € O(n) 9. T(n) = n³ + 20n € №(n² ) 10. T(n) = 2n³ - 7n + 1 € ☺(n³)

NE NFA = {N| N is an NFA and L(N) = Ø}. How to show it is in NP

Let P(n) be the statement “the number 2^2n -1 is divisible by 3”. Use mathematicalinduction to show that P(n) is true for all n >0