2. Using the digits 0 and 5, write all ten-digit numbers that are divisible by 9. How many such numbers are there?
2. The sum of the digits of any ten-digit number composed of the digits 0 and 5 is
a multiple of the number 5.
2 POINTS
Furthermore, since we are looking for numbers that are divisible by 9, the sum of the digits must be divisible by 9.
Since the numbers 5 and 9 are relatively prime, it follows that the desired numbers must have exactly nine fives and one zero. Then the sum of the digits is $9 \cdot 5 = 45$.
2 POINTS
Zero cannot be in the first position, so we have the numbers: $5055555555, 5505555555, 5550555555$,
$5555055555, 5555505555, 5555550555, 5555555055, 5555555505, 55555555550$.
5 POINTS
Thus, there are a total of 9 numbers that satisfy the conditions of the problem.
1 POINT
TOTAL 10 POINTS