MP Board · Class 10 · Mathematics · Real NumbersExplain why the number $4^n$ cannot end with the digit zero for any natural number $n$.

For any number to end with the digit zero, its prime factorization must contain both 2 and 5 as prime factors. The prime factorization of $4^n$ is $(2^2)^n = 2^{2n}$. It is clearly observed that 5 does not occur in the prime factorization of $4^n$. Therefore, by the uniqueness of the Fundamental Theorem of Arithmetic, $4^n$ can never end with the digit zero for any natural number $n$.