There are 251 numbers from 1000 to 2000, inclusive, that are divisible by 4.
Here's how we determine that:
- Understanding Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- First Multiple: The first number in the range 1000 to 2000 that is divisible by 4 is 1000 (1000/4=250).
- Last Multiple: The last number in the range that is divisible by 4 is 2000 (2000/4 = 500).
- Arithmetic Sequence: The multiples of 4 form an arithmetic sequence: 1000, 1004, 1008... 2000. We can use the formula to find the number of terms in this sequence.
- Formula: Let the first term be 'a', the last term be 'l', and the common difference be 'd'. Then the number of terms 'n' is: n = ((l - a) / d) + 1
- Applying the formula:
- a = 1000
- l = 2000
- d = 4
- n = ((2000 - 1000) / 4) + 1
- n = (1000 / 4) + 1
- n = 250 + 1
- n = 251
Reference Analysis
The provided reference states: "So n = 1000 and X is 4. So 250. If we are excluding the endpoints then subtract 1 because 2000 is divisible by 4 so, 249". This portion seems to be addressing the *number of multiples of 4 between 1000 and 2000 when endpoints are excluded. However, the question asks for numbers from 1000 to 2000, so the endpoints are included. Based on the correct calculation as demonstrated above, there are 251 numbers.
Therefore, there are 251 numbers between 1000 and 2000 (inclusive) that are divisible by 4.
