n³ – 7n + 3 is divisible by 3, for all natural numbers n.

Given; P(n) = n³ – 7n + 3 is divisible by 3.

P(0) = 0³ – 7×0 + 3 = 3; is divisible by 3.

P(1) = 1³ – 7×1 + 3 = −3; is divisible by 3.

P(2) = 2³ – 7×2 + 3 = −3; is divisible by 3.

P(3) = 3³ – 7×3 + 3 = 9; is divisible by 3.

Let P(k) = k³ – 7k + 3 is divisible by 3; ⇒ k³ – 7k + 3 = 3x.

⇒ P(k+1) = (k+1)³ – 7(k+1) + 3

= k³ + 3k² + 3k + 1 – 7k – 7 + 3

= 3x + 3(k² + k – 2); is divisible by 3.

⇒ P(k+1) is true when P(k) is true.

∴ By Mathematical Induction P(n) = n³ – 7n + 3 is divisible by 3, for all natural numbers n.