Question

For any positive integer m ______ is divisible by 4.

(a) 5m^2 + 2

(b) 3m + 1

(c) m^2 + 3

(d) m^3 + 3m

I had been asked this question during an interview.

I want to ask this question from Principle of Mathematical Induction topic in division Induction and Recursion of Discrete Mathematics

Answer 1

Right answer is (d) m^3 + 3m

Best explanation: The required answer is, m^3 + 3m. Now, by induction hypothesis, we have to prove for m=k, k^3+3k is divisible by 4. So, (k + 1)^3 + 3 (k + 1) = k^3 + 3 k^2 + 6 k + 4

= [k^3 + 3 k] + [3 k^2 + 3 k + 4] = 4M + (12k^2 + 12k) – (8k^2 + 8k – 4), both the terms are divisible by 4. Hence (k + 1)^3 + 3 (k + 1) is also divisible by 4 and hence it is proved for any integer m.