Final answer:
We can use mathematical induction to prove that 8ⁿ-3ⁿ is divisible by 5 for all natural numbers n.
Step-by-step explanation:
To show that 8ⁿ-3ⁿ is divisible by 5 for all natural numbers n, we can use mathematical induction.
Step 1: Show that the statement is true for n = 1:
8¹-3¹ = 8-3 = 5, which is divisible by 5.
Step 2: Assume the statement is true for some arbitrary integer k:
8ᵏ-3ᵏ is divisible by 5.
Step 3: Show that the statement is true for k+1:
8^(k+1)-3^(k+1) = (8ᵏ*8) - (3ᵏ*3) = 8ᵏ*8 - 3ᵏ*3 + 5 - 5 = (8ᵏ-3ᵏ) * 8 + 5 - 5.
Since 8ᵏ-3ᵏ is divisible by 5 (per our assumption in Step 2), we can rewrite the equation as (divisible by 5) * 8 + 5 - 5. Since 8 times any integer is divisible by 5, the entire expression is divisible by 5.
Therefore, by mathematical induction, 8ⁿ-3ⁿ is divisible by 5 for all natural numbers n.