Question

Show that 8^n can never end with digit 0 for any natural number n.

Answer 1

for ending the no. with zero the no. should have prime factorisation of 2^n×5^m.

but since 8^n has prime factorisation of 2×2×2×ntimes (of 8) so this no. 8^n can never end with zero.

Answer 2

 let nₓ = 8ˣ
 let x=1
n₁ = 8
8 is not divisible by 10 → hence true for x=1

let nₐ = 8ᵃ  is not divisible by 10
∴ 8ᵃ = 10N+K where  K ∈ N & K≠0 and  0<K<10 and k is even

consider nₐ₊₁ = 8ᵃ⁺¹ = 8.8ᵃ = (10-2)8ᵃ = 10.8ᵃ - 2.8ᵃ 
10.8ᵃ - 2.8ᵃ  is divisible by 10 if and only if both 10.8ᵃ, 2.8ᵃ are both divisible by 10
10.8ᵃ is divisible by 10....................................1
consider 2.8ᵃ = 2(10N+K) = 20N+2K
2K is not divisible 10 as K is even and hence not divisible by 5
∴nₐ₊₁ is not divisible by 10  when nₐ is not divisible by 10

 hence by induction 8ˣ is not divisible by 10