Final answer:
To find the number of three-digit numbers that can be formed from the set (0,1,2,...9) with repetition allowed and the number must be even, there are 500 possible numbers. There are 200 numbers that are divisible by 5, 100 numbers that are divisible by 10, and 100 numbers that are divisible by 25.
Step-by-step explanation:
To find the number of three-digit numbers that can be formed from the set (0,1,2,...9) with repetition allowed and the number must be even, we need to consider the possible values for each digit. The first digit can be any number from the set (0,2,4,6,8) which gives us 5 options. The second and third digits can be any number from the set (0,1,2,...9), so they each have 10 options. Therefore, the total number of three-digit even numbers that can be formed is 5 * 10 * 10 = 500.
To determine the number of three-digit numbers that are divisible by 5, we need to consider the possible values for each digit. The first two digits can be any number from the set (0,1,2,...9) which gives us 10 options each. The last digit must be either 0 or 5, so it has 2 options. Therefore, the total number of three-digit numbers divisible by 5 is 10 * 10 * 2 = 200.
To find the number of three-digit numbers that are divisible by 10, we need to consider the possible values for each digit. The first two digits can be any number from the set (0,1,2,...9) which gives us 10 options each. The last digit must be 0, so it has 1 option. Therefore, the total number of three-digit numbers divisible by 10 is 10 * 10 * 1 = 100.
To determine the number of three-digit numbers that are divisible by 25, we need to consider the possible values for each digit. The first two digits can be any number from the set (0,1,2,...9) which gives us 10 options each. The last two digits must be 25, so they have 1 option each. Therefore, the total number of three-digit numbers divisible by 25 is 10 * 10 * 1 = 100.