Question

Show that 6ⁿ can never end with 0 for any natural number n.

Answer 1

Given:

If the statement means 6n (six times n): the claim “6n can never end with 0” is ambiguous/false.

If the statement means 6ⁿ (six to the power n): the claim “6ⁿ can never end with 0” is true.

Case 1: 6n (product)

1. “Ends with 0” ⇔ divisible by 10 ⇔ divisible by 2 and 5.

2. 6n is always divisible by 2 (since 6 contains a factor 2). For 6n to be divisible by 5 we need 5 n.

3. Take n = 5 (a natural number).

Then 6 × 5 = 30, which ends with 0. 

The statement is false for 6n multiples of 6 that have n a multiple of 5 do end with 0 (e.g. n = 5).

Case 2: 6ⁿ (power)

1. 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ, so the only prime factors of 6ⁿ are 2 and 3.

2. For a number to end with 0, it must be divisible by 10, hence must have 5 as a prime factor.

3. 6ⁿ has no factor 5, so it cannot be divisible by 10 and therefore cannot end with 0.

6ⁿ can never end with 0 for any natural number n.