There are exactly 300 numbers divisible by 3 between 100 and 1000.
The question asks us to find the count of numbers between 100 and 1000 that are divisible by 3. According to the provided reference, there are 901 integers from 100 to 1000, or 899 integers between 100 and <1000. Crucially, neither 100 nor 1000 are divisible by 3.
We know that the first number greater than 100 divisible by 3 is 102 (3 34), and the last number less than 1000 divisible by 3 is 999 (3 333). To find how many multiples of 3 exist between 102 and 999, inclusive, we can use the following logic:
- Identify the first multiple of 3 in the range: 102
- Identify the last multiple of 3 in the range: 999
- Calculate the total count: According to the reference, starting from 102 up to 999, every third number is divisible by 3. So, there are (999-102) = 897 numbers, divide by 3 is 897/3 = 299. Then add 1 to include the first number 102, that is 299 + 1 = 300 numbers.
Here's another way to think about it:
- The first multiple of 3 is 102 which can be expressed as 3 * 34.
- The last multiple of 3 is 999 which can be expressed as 3 * 333.
The count of numbers divisible by 3 is therefore the number of integers from 34 to 333, inclusive, which is 333 - 34 + 1 = 300.
| Range | First Multiple of 3 | Last Multiple of 3 | Calculation | Result |
|---|---|---|---|---|
| 100 to 1000 | 102 | 999 | (999-102)/3 + 1 | 300 |
| 3 * 34 | 3 * 333 | 333 - 34 + 1 | 300 |
Therefore, there are 300 numbers between 100 and 1000 that are divisible by 3.
