Explanation:
We can prove the statement by mathematical induction.
Base Case: For n = 0, we have 9n - 3n = 1, which is divisible by 6, since 1 = 6*0 + 1.
Inductive Step: Assume that 6 divides 9k - 3k for some non-negative integer k. We need to show that 6 also divides 9(k+1) - 3(k+1).
Starting with 9(k+1) - 3(k+1), we can simplify it as follows:
9(k+1) - 3(k+1) = 9k + 9 - 3k - 3
= (9k - 3k) + (9 - 3)
= 6k + 6
Since 6 divides both 6k and 6, it also divides their sum, 6k + 6. Therefore, we have shown that 6 divides 9(k+1) - 3(k+1).
By the principle of mathematical induction, we can conclude that 6 divides 9n - 3n for all non-negative integers n.