Question

Show that any number of the form 4ⁿ, n ∈ N can never end with the digit 0.

Answer 1

Given: Let the number be 4ⁿ, where n ∈ N. We want to show 4ⁿ cannot end with the digit 0.

Step-wise calculation:

1. If an integer ends with digit 0, it is divisible by 10, so it must have both prime factors 2 and 5.

2. 4ⁿ = (2²)ⁿ = 2²ⁿ, so the prime factorization of 4ⁿ contains only the prime 2 (no factor 5).

3. By uniqueness of prime factorization, 4ⁿ cannot have 5 as a factor, hence it cannot be divisible by 10.

Therefore, no number of the form 4ⁿ (n ∈ N) can end with the digit 0.