To determine how many multiples of 3 exist between 10 and 787, follow these steps:
1. Identify the smallest multiple of 3 greater than or equal to 10:
- Start with the lower bound, which is 10.
- The smallest multiple of 3 greater than or equal to 10 can be found by checking the closest multiple of 3 above 10.
- Since 10 is not a multiple of 3, you need to go up to the next number that is a multiple of 3.
- By inspection, you get that 12 is the first multiple of 3 that is greater than or equal to 10.
2. Identify the largest multiple of 3 less than or equal to 787:
- Start with the upper bound, which is 787.
- The largest multiple of 3 less than or equal to 787 can be found by checking the closest multiple of 3 below 787.
- Since 787 is not a multiple of 3, you need to go down to the previous number that is a multiple of 3.
- By inspection, you see that 786 is the largest multiple of 3 that is less than or equal to 787.
3. Calculate the number of multiples of 3 between 12 and 786 inclusive:
- Each multiple of 3 can be represented as 3n, where n is an integer.
- Find the sequence of integers n corresponding to the given range.
- For the first multiple (12 = 3 4), n = 4.
- For the last multiple (786 = 3 262), n = 262.
- The number of multiples is the count of integers from 4 to 262 inclusive.
4. Calculate the total count:
- The total count is the difference between the last and first n values, plus one.
- This yields (262 - 4) + 1.
- Performing the calculation provides:
[tex]\[
(262 - 4) + 1 = 259
\][/tex]
Therefore, there are 259 multiples of 3 between 10 and 787.
