How many positive five-digit whole numbers are there for which the product of their five digits is 900?

(Karl Czakler)

Answer. 210
It is given that $900=30^{2}=5^{2} \cdot 3^{2} \cdot 2^{2}$. Therefore, the sought number must contain the digit 5 twice. The product of the other three digits is then 36.

Therefore, the following possibilities exist for the five digits of the sought number:

(a) $5,5,9,4,1$

(b) $5,5,9,2,2$

(c) $5,5,6,6,1$

(d) $5,5,6,3,2$

(e) $5,5,4,3,3$

There are 10 possibilities for the distribution of the two digits 5 in a five-digit number.

If the other three digits are different, there are 6 possibilities each; if two of the other three digits are the same, there are only 3 different arrangements for these three digits. Therefore, we have

$$
10 \cdot 6 + 10 \cdot 3 + 10 \cdot 3 + 10 \cdot 6 + 10 \cdot 3 = 210
$$

numbers with this property.

(Karl Czakler)