Understanding the Problem
To find the sum of all natural numbers up to 1000 that are neither divisible by 2 nor by 5, we first need to identify the numbers that meet this condition.
Identifying Eligible Numbers
- Natural Numbers: These are the counting numbers starting from 1.
- Divisibility Conditions: A number is excluded if it is divisible by:
- 2 (even numbers)
- 5 (multiples of 5)
Finding the Range
- The natural numbers we are considering range from 1 to 1000.
Finding Eligible Numbers
- The eligible numbers are those that are:
- Odd (not divisible by 2)
- Not ending in 0 or 5 (not divisible by 5)
This leaves us with numbers that are odd and end with 1, 3, 7, or 9.
Counting Eligible Numbers
- Total odd numbers up to 1000: There are 500 odd numbers (1, 3, 5, ..., 999).
- Odd numbers divisible by 5: These are 5, 15, 25, ..., 995. The largest odd multiple of 5 less than 1000 is 995, which gives us 100 terms (5n where n=1 to 100).
- Eligible Odd Numbers Count: 500 (total odd numbers) - 100 (odd multiples of 5) = 400 eligible numbers.
Calculating the Sum
To find the sum of these eligible numbers:
- Consider the arithmetic sequence: 1, 3, 7, 9, ..., up to 999.
- The sum of an arithmetic series can be calculated using the formula:
- Sum = n/2 * (first term + last term)
- Here, n = 400, first term = 1, last term = 999.
The final sum gives us the desired total of all natural numbers up to 1000 that are neither divisible by 2 nor by 5.

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