Find how many positive integers, less than 1000, are

1. Identify the sequence of odd numbers less than 1000 and express it as an arithmetic progression.

The positive odd integers less than 1000 form an arithmetic progression: $1, 3, 5, \dots, 999$.
Here, the first term $a = 1$ and the common difference $d = 2$.

2. Use the formula for the $n$-th term to find the number of odd numbers.

Let the last term be the $n$-th term, $T_n = 999$.
$a + (n - 1)d = 999$
$1 + (n - 1)2 = 999$
$2(n - 1) = 998$
$n - 1 = 499$
$n = 500$

1. Find the total number of odd positive integers less than 1000 (re-deriving part i).

The sequence of odd positive integers less than $1000$ is $1, 3, 5, \dots, 999$.
Using the formula for the $n$th term of an arithmetic progression, $u_n = a + (n-1)d$:
$999 = 1 + (n-1)(2)$
$998 = 2(n-1) \implies n = 500$
There are $500$ odd positive integers.

2. Find the number of odd positive integers less than 1000 that are divisible by 5.

Odd numbers that are divisible by $5$ must end with the digit $5$. The sequence is $5, 15, 25, \dots, 995$.
Using $u_k = a + (k-1)d$:
$995 = 5 + (k-1)(10)$
$990 = 10(k-1) \implies k = 100$
There are $100$ odd numbers divisible by $5$.

3. Calculate the number of odd positive integers not divisible by 5.

Subtract the number of odd integers divisible by $5$ from the total number of odd integers:
$500 - 100 = 400$

1. Calculate the sum of all positive odd integers less than 1000.

Let $S_1$ be the sum of all positive odd integers less than $1000$.
This sequence is $1, 3, 5, \dots, 999$, which is an arithmetic progression with first term $a = 1$, common difference $d = 2$, and last term $l = 999$.
The number of terms is $n = \frac{999-1}{2} + 1 = 500$.
$S_1 = \frac{n}{2}(a + l) = \frac{500}{2}(1 + 999) = 250000$

2. Calculate the sum of positive odd integers less than 1000 that are divisible by 5.

Let $S_2$ be the sum of positive odd integers less than $1000$ that are divisible by $5$.
These numbers must end in $5$ to be odd, forming the sequence $5, 15, 25, \dots, 995$.
This is an arithmetic progression with first term $a = 5$, common difference $d = 10$, and last term $l = 995$.
The number of terms is $m = \frac{995 - 5}{10} + 1 = 100$.
$S_2 = \frac{m}{2}(a + l) = \frac{100}{2}(5 + 995) = 50000$

3. Subtract the sum of the odd multiples of 5 from the total sum of odd integers.

The required sum is the total sum of odd integers minus the sum of those divisible by $5$:
$\text{Required sum} = S_1 - S_2 = 250000 - 50000 = 200000$