## 1. How many numbers exactly have 3 digits?

The question of how many numbers exactly have 3 digits can be answered by considering the number range from 100 to 999. In this range, the first three-digit number is 100, and the last one is 999. To determine the count of these numbers, we can subtract the first three-digit number from the last and then add one. Therefore, the total number of three-digit numbers is 999 – 100 + 1 = 900.

## 2. What is the range of three-digit numbers?

The range of three-digit numbers starts from the smallest three-digit number, which is 100, and ends at the largest three-digit number, which is 999. Therefore, the range of three-digit numbers spans from 100 to 999.

## 3. How many even three-digit numbers are there?

To determine the count of even three-digit numbers, we need to consider that even numbers always end with 0, 2, 4, 6, or 8. Since the first digit of a three-digit number can range from 1 to 9 (excluding 0), the second and third digits can take any number from 0 to 9. Therefore, there are 5 even digits for the units place, 10 options for the tens place, and 9 options for the hundreds place (excluding 0). Consequently, the total count of even three-digit numbers is 5 x 10 x 9 = 450.

## 4. How many odd three-digit numbers are there?

Odd three-digit numbers are those that do not end with 0, 2, 4, 6, or 8. Since the first digit of a three-digit number can range from 1 to 9 (excluding 0), the second and third digits can take any number from 0 to 9. Therefore, there are 5 odd digits for the units place (1, 3, 5, 7, 9), 10 options for the tens place, and 9 options for the hundreds place (excluding 0). Consequently, the total count of odd three-digit numbers is 5 x 10 x 9 = 450.

## 5. How many three-digit numbers are divisible by 3?

To determine the count of three-digit numbers divisible by 3, we need to find the number of multiples of 3 within the range of three-digit numbers (100 to 999). The smallest three-digit multiple of 3 is 102, and the largest is 999. By subtracting 102 from 999 and then dividing the result by 3, we can obtain the count. Therefore, (999 – 102) / 3 + 1 = 299 – 34 + 1 = 266.

## 6. How many three-digit numbers are divisible by 7?

To find the count of three-digit numbers divisible by 7, we need to determine the number of multiples of 7 within the range of three-digit numbers (100 to 999). The smallest three-digit multiple of 7 is 105, and the largest is 994. By subtracting 105 from 994 and then dividing the result by 7, we can obtain the count. Therefore, (994 – 105) / 7 + 1 = 889 / 7 + 1 = 127 + 1 = 128.

## 7. How many three-digit numbers have a sum of digits equal to 15?

To determine the count of three-digit numbers with a sum of digits equal to 15, we can start by listing down the possible combinations of three digits that sum up to 15. These combinations are:

– 6, 5, 4

– 7, 4, 4

– 7, 5, 3

– 8, 3, 4

– 8, 4, 3

– 8, 5, 2

– 9, 2, 4

– 9, 3, 3

– 9, 4, 2

– 9, 5, 1

– 9, 1, 5

– 8, 6, 1

– 8, 1, 6

– 7, 7, 1

– 7, 1, 7

– 6, 6, 3

– 6, 3, 6

– 5, 5, 5

By counting these combinations, we find that there are 18 three-digit numbers with a sum of digits equal to 15.

## 8. How many three-digit numbers are prime?

To determine the count of three-digit prime numbers, we need to find prime numbers within the range of three-digit numbers (100 to 999). By checking each number individually, we can identify the primes. The prime numbers in this range are: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.

By counting these numbers, we find that there are 143 three-digit prime numbers.

## 9. How many three-digit numbers are divisible by both 2 and 5?

To determine the count of three-digit numbers divisible by both 2 and 5, we need to find the common multiples of 2 and 5 within the range of three-digit numbers (100 to 999). Since the numbers divisible by 5 must end with 0 or 5, and the numbers divisible by 2 must end with an even digit (0, 2, 4, 6, or 8), the only common digit for the units place is 0. Therefore, the count of three-digit numbers divisible by both 2 and 5 is 1 (ending with 0).

## 10. How many three-digit numbers have a repeating digit?

To find the count of three-digit numbers with a repeating digit, we can consider the digits in each place (hundreds, tens, and units) separately. For a repeating digit in the hundreds place, it can take any value from 1 to 9, resulting in 9 options. The tens and units digits can have any value from 0 to 9, including the repeating digit, giving them 10 options each. Therefore, the total count is 9 x 10 x 10 = 900.

## 11. How many three-digit numbers have all unique digits?

To determine the count of three-digit numbers with all unique digits, we need to consider the digits in each place (hundreds, tens, and units) separately. For the hundreds place, there are 9 options (excluding 0 and the value already used). The tens place can have any value from 0 to 9, except for the digit already used. Therefore, there are 9 options for the tens place. Finally, the units place can have any value from 0 to 9, except for the two digits already used. So, there are 8 options for the units place. Consequently, the total count is 9 x 9 x 8 = 648.

## 12. How many three-digit numbers have two equal digits?

To find the count of three-digit numbers with two equal digits, we can consider the cases where the two equal digits are in the tens and units places, and the cases where the two equal digits are in the hundreds and tens places. For the tens and units place scenario, we have 10 options for the repeated digit (0 to 9) and 9 options for the remaining digit. Therefore, there are 10 x 9 = 90 possibilities. Similarly, for the hundreds and tens place scenario, we have 9 options for the repeated digit (excluding 0) and 10 options for the remaining digit. Thus, there are 9 x 10 = 90 possibilities. Adding these two scenarios together, we get a total count of 90 + 90 = 180.

## 13. How many three-digit numbers have digits in descending order?

To determine the count of three-digit numbers with digits in descending order, we can consider the digits in each place (hundreds, tens, and units) separately. For the hundreds place, there are 8 options (excluding 0 and the value already used). The tens place can have any value from 0 to the previous digit, resulting in the same number of options as the hundreds place. Finally, the units place can have any value from 0 to the previous digit, giving it a maximum of 9 options. Therefore, the total count is 9 x 8 x 7 = 504.

## 14. How many three-digit numbers have digits in ascending order?

To find the count of three-digit numbers with digits in ascending order, we can consider the digits in each place (hundreds, tens, and units) separately. For the hundreds place, there are 1 option (0). The tens place can have any value from the previous digit to 9, resulting in 9 options. Finally, the units place can have any value from the previous digit to 9, giving it the same number of options as the previous digit. Therefore, the total count is 1 x 9 x 9 = 81.

## 15. How many three-digit numbers have digits in non-ascending order?

To determine the count of three-digit numbers with digits in non-ascending order, we can consider the digits in each place (hundreds, tens, and units) separately. For the hundreds place, there are 10 options (0 to 9). The tens place can have any value from 0 to the previous digit, resulting in the same number of options as the hundreds place. Finally, the units place can have any value from 0 to the previous digit, giving it a maximum of 10 options. Therefore, the total count is 10 x 10 x 10 = 1000.

## 16. How many three-digit numbers have digits in non-descending order?

To find the count of three-digit numbers with digits in non-descending order, we can consider the digits in each place (hundreds, tens, and units) separately. For the hundreds place, there are 9 options (excluding 0 and the value already used). The tens place can have any value from the previous digit to 9, resulting in the same number of options as the hundreds place. Finally, the units place can have any value from the previous digit to 9, giving it the same number of options. Therefore, the total count is 9 x 9 x 9 = 729.

## 17. How many three-digit numbers have consecutive digits?

To determine the count of three-digit numbers with consecutive digits, we can consider the cases where the three digits are consecutive and in ascending order, and the cases where the three digits are consecutive and in descending order. For the ascending order scenario, the repeatable patterns are 012, 123, 234, 345, 456, 567, 678, and 789, while the descending order scenario includes 987, 876, 765, 654, 543, 432, 321, and 210. Therefore, there are 8 patterns with consecutive digits in both ascending and descending order.

## 18. How many three-digit numbers have a digit sum of 18?

To determine the count of three-digit numbers with a digit sum of 18, we can use a similar approach as the case of a sum of 15 digits. By listing down the possible combinations of three digits with a sum of 18, we find the following patterns:

– 9, 9, 0

– 9, 8, 1

– 9, 7, 2

– 9, 6, 3

– 9, 5, 4

– 8, 8, 2

– 8, 7, 3

– 8, 6, 4

– 8, 5, 5

– 7, 7, 4

– 7, 6, 5

– 6, 6, 6

By counting these combinations, we find that there are 12 three-digit numbers with a digit sum of 18.

## 19. How many three-digit numbers have a digit sum greater than 20?

To determine the count of three-digit numbers with a digit sum greater than 20, we need to list the possible combinations of digits that sum up to values greater than 20. Considering that the maximum digit sum in three digits is 27 (9 + 9 + 9), any combination that yields a sum greater than 20 is acceptable. By listing down the possible combinations, we find the following patterns:

– 9, 9, 3

– 9, 9, 4

– 9, 9, 5

– 9, 9, 6

– 9, 9, 7

– 9, 9, 8

– 9, 8, 4

– 9, 8, 5

– 9, 8, 6

– 9, 8, 7

– 9, 7, 5

– 9, 7, 6

– 9, 7, 7

– 9, 6, 6

– 8, 8, 5

– 8, 8, 6

– 8, 8, 7

– 8, 7, 7

By counting these combinations, we find that there are 17 three-digit numbers with a digit sum greater than 20.

## 20. How many three-digit numbers have a digit sum less than 10?

To determine the count of three-digit numbers with a digit sum less than 10, we need to list the possible combinations of digits that sum up to values less than 10. By considering various combinations, we find the following patterns:

– 1, 1, 1

– 1, 1, 2

– 1, 1, 3

– 1, 1, 4

– 1, 1, 5

– 1, 1, 6

– 1, 1, 7

– 1, 1, 8

– 1, 2, 1

– …

– 2, 2, 5

– 2, 2, 6

– 2, 2, 7

– 2, 3, 1

– 2, 3, 2

– …

– 7, 1, 1

– 7, 1, 2

– …

– 9, 1, 1

By counting these combinations, we find that there are 45 three-digit numbers with a digit sum less than 10.

## 21. How many three-digit numbers have the digit 0?

To determine the count of three-digit numbers with the digit 0, we can consider the cases where 0 is present in the units, tens, or hundreds place. For the units place scenario, there are 9 options for the hundreds digit (excluding 0) and 10 options for the tens digit (including 0). Therefore, there are 9 x 10 = 90 possibilities. Similarly, for the tens place scenario, there are 9 options for the hundreds digit (excluding 0) and 9 options for the units digit (excluding 0). Thus, there are 9 x 9 = 81 possibilities. Finally, for the hundreds place scenario, there are 8 options for the tens digit (excluding 0) and 9 options for the units digit (excluding 0). Consequently, there are 8 x 9 = 72 possibilities. Adding these three scenarios together, we get a total count of 90 + 81 + 72 = 243.

## 22. How many three-digit numbers have all odd digits?

To determine the count of three-digit numbers with all odd digits, we need to consider three odd digits: 1, 3, and 5. For the hundreds place, any of the three odd digits can be used, resulting in 3 options. Similarly, for the tens and units places, there are also 3 options each. Therefore, the total count is 3 x 3 x 3 = 27.