There are 1000 numbers between 1000 and 9999 that are divisible by 9.
The question asks for the count of numbers within a specific range (1000 to 9999) that are divisible by 9. Here's how we can understand this:
-
Divisibility Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
-
Range: We're considering only numbers between 1000 and 9999, which are four-digit numbers.
-
Finding the First Number: The first number in this range divisible by 9 is 1008 (1 + 0 + 0 + 8 = 9, which is divisible by 9).
-
Finding the Last Number: The last number in this range divisible by 9 is 9999 (9 + 9 + 9 + 9 = 36, which is divisible by 9).
We could find all the numbers in this range that are divisible by 9 and count them to get 1000. Alternatively we can use an arithmetic sequence to solve this:
- The first term (a) is 1008.
- The common difference (d) is 9.
- The last term (l) is 9999.
If the total number of terms is denoted by 'n', then the last term can be written as:
l = a + (n-1) d
9999 = 1008 + (n-1) 9
Solving for n:
9999 - 1008 = (n-1) 9
8991 = (n-1) 9
8991/9 = n -1
999 = n -1
n = 1000
Therefore, there are 1000 numbers between 1000 and 9999 that are divisible by 9.
