Problem 2. Determine the smallest natural number a such that $378 \cdot a=b \cdot b$, where $b$ is a natural number.
Solution. For the product 378.a to be equal to the product of two identical natural numbers, the product 378$\cdot$a must have an even number of identical prime factors. From $378=2 \cdot 3 \cdot 3 \cdot 3 \cdot 7$ it follows that 2, 3, and 7 are prime factors of the number $a$. Therefore, $a=2 \cdot 3 \cdot 7$, i.e., $a=42$. According to this, the smallest natural number with the desired property is the number 42.