Question

Show that 12ⁿ cannot end with the digit 0 or 5 for any natural number n.

Answer 1

If any number ends with the digit 0 or 5, it is always divisible by 5.

If 12ⁿ ends with the digit zero or five it must be divisible by 5.

This is possible only if prime factorisation of 12ⁿ contains the prime number 5.

Now, 12 = 2 × 2 × 3 = 2² × 3

12ⁿ = (2² × 3)ⁿ = 2²ⁿ × 3ⁿ

Since, there is no term containing 5.

Hence, there is no value of n ∈ N for which 12ⁿ ends with the digit zero or five.