45. Consider the number 12ⁿ, where n is a natural number. Check whether there is any value of n ∊ N for which 12ⁿ ends with the digit zero.

We know, if any number ends with the digit zero it is always divisible by 5.

⇒ If 12ⁿ ends with the digit zero, 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 = 2² × 3ⁿ

i.e., prime factorisation of 12ⁿ does not contain the prime number 5.

⇒ There is no value of n ∊ N for which 12ⁿ ends with the digit zero.