## Task A-1.7.

How many five-digit natural numbers are there whose product of digits is equal to 900?
## Solution.

The prime factorization of the number 900 is $900=2^{2} \cdot 3^{2} \cdot 5^{2}$. Among the non-zero decimal digits, only the digit 5 is divisible by 5, so exactly two of the digits in the five-digit number must be 5.

There are $\frac{5 \cdot 4}{2}=10$ ways to place the two 5s in some of the five positions in the five-digit number.

In the five-digit number, we now need to determine the remaining three digits that multiply to $2^{2} \cdot 3^{2}=36$ and arrange them in the three remaining positions.

Notice that $2^{2} \cdot 3^{2}=36$ can be factored into three digits in 5 ways:

$$
1 \cdot 4 \cdot 9, \quad 2 \cdot 3 \cdot 6, \quad 1 \cdot 6 \cdot 6, \quad 2 \cdot 2 \cdot 9, \quad 3 \cdot 3 \cdot 4 . \quad 1 \text { point }
$$

In the first two cases, all three digits are different, so there are $3 \cdot 2 \cdot 1=6$ different ways to arrange these three digits in the remaining positions.

In the last three cases, we have three digits where two are the same. Determining one of the three positions for the unique digit uniquely determines the positions of the remaining (identical) digits. Therefore, in these cases, there are 3 different arrangements of these digits.

Thus, we conclude that the total number of five-digit numbers with the desired property is

$$
10 \cdot(2+2+3+3+3)=210
$$

Note: Each of the 5 cases in the above solution is worth 1 point, corresponding to the last five points in the above scoring scheme.

If students do not consider the digits 5 separately but reduce the problem to the cases

$1 \cdot 4 \cdot 9 \cdot 5 \cdot 5, \quad 2 \cdot 3 \cdot 6 \cdot 5 \cdot 5, \quad 1 \cdot 6 \cdot 6 \cdot 5 \cdot 5, \quad 2 \cdot 2 \cdot 9 \cdot 5 \cdot 5, \quad 3 \cdot 3 \cdot 4 \cdot 5 \cdot 5$, each of these cases is worth 2 points.

## SCHOOL COMPETITION IN MATHEMATICS

## 2nd grade - high school - A variant

## January 26, 2024.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO SCORE AND EVALUATE THAT APPROACH APPROPRIATELY.