Can the number ($\displaystyle 15$) $\displaystyle { }^{\mathrm{n}}$, n being a natural number, end with the digit $\displaystyle 0$ ? Give reasons.

\(\displaystyle \mathbf{1 5}^{\mathbf{n}}=\mathbf{5}^{\mathbf{n}} \times \mathbf{3}^{\mathbf{n}}\) A number ends with zero if it has two prime factors $\displaystyle 2$ and $\displaystyle 5$ both. Since \(\displaystyle \mathbf{1 5}^{\mathbf{n}}\) does not have $\displaystyle 2$ as a prime factor, so it can't end with zero