3B. Determine the smallest natural number, the product of whose digits is equal to 75600.
Solution. We factorize the number 75600 into prime factors and get $75600=2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7$. Since all prime divisors of 75600 are single-digit numbers, this means there exists a number such that the product of its digits is equal to 75600.

The desired number contains the digits 5, 5, and 7, because multiplying any of the numbers 2, 3, 5, 7 by 5 or 7 results in a number greater than 9. The product $2^{4} 3^{3}$ can be written with at most 7 digits ($2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3$), and with at least 3 digits (6$\cdot$8.9). Since we are looking for the smallest natural number, we conclude that it is written with the digits $6,8,9,5,5,7$, which is the number 556789.