1. The smallest natural number whose square ends with three fours is 38, since $38^{2}=1444$. What is the next smallest natural number with this property?
I/1. Let $38+n$ be the sought number. Then $(38+n)^{2}=1444+n(76+n)$, where the number $n(76+n)$ ends with three zeros, or is a multiple of 1000. Since $1000=5^{3} \cdot 2^{3}$, either $n$ or $76+n$ must be divisible by 5. However, $n$ and $76+n$ cannot both be divisible by 5, so one of these two numbers must be divisible by $125=5^{3}$ and thus of the form $125 k$.

Let's also consider the factor $2^{3}$. From $76=2^{2} \cdot 19$, it follows that the numbers $n$ and $76+n$ have the same remainder when divided by 4. Since the product $n(76+n)$ must be divisible by 8, at least one of the numbers $n$ or $76+n$ must be divisible by 4. Therefore, both are divisible by 4, and one of them is of the form $125 \cdot 4 \cdot m=500 \mathrm{~m}$, where $m \in \mathbb{N}$. To find the smallest possible value, we choose $m=1$ and from $76+n=500$ we calculate $38+n=462$.