There are 33 numbers between 1 and 1000 that are divisible by 15 but not by 30. Here's how we arrive at that answer:
-
Numbers divisible by 15: A number is divisible by 15 if it can be written in the form 15*n, where n is an integer.
-
Numbers divisible by 30: A number is divisible by 30 if it can be written in the form 30m, where m is an integer. Notice that any number divisible by 30 is also* divisible by 15 (since 30 is a multiple of 15).
-
The Key Insight: A number divisible by 15 but not by 30 must have the form 15 (odd number). This is because 15 (even number) will always be divisible by 30. In other words, we are looking for multiples of 15 that, when divided by 2, leave a remainder.
-
Finding the numbers:
-
The smallest number divisible by 15 is 15 (15 * 1).
-
The largest number less than or equal to 1000 that is divisible by 15 is 990 (15 * 66). Note: 1000 / 15 ≈ 66.67.
-
So, we need to count how many odd numbers there are between 1 and 66 (inclusive). These odd numbers will be the 'n' in the 15*n form to give us a number divisible by 15 but not by 30.
-
We want the number of odd integers n such that 1 ≤ n ≤ 66. The odd integers are 1, 3, 5, ..., 65.
-
To find how many odd numbers there are between 1 and 66, we can use the formula: (Last odd number - First odd number)/2 + 1. So (65 - 1)/2 + 1 = 64/2 + 1 = 32 + 1 = 33.
-
Therefore, there are 33 numbers between 1 and 1000 that are divisible by 15 but not by 30. These numbers are 15, 45, 75, ..., 975.
