c

\(1000 = 2^3*5^3\)

So basically all even integers are automatically eliminated

There are 500 odd integers from 0 to 1000

Now we also have 5 as a factor. Multiples of 5 have either 0 or 5 at their one's digit place.
The ones with 0 at one's digit place are already eliminated since they are even.

We have to eliminate all integers that have 5 at their one's digit place. There are 100 such numbers.

Therefore, \(N = 500 - 100 = 400\)

But, the question mentions only integer, there is nothing given about it needing to be positive or negative. Even if you don't consider negative number, 0 satisfies the questions requirements and now \(N\) = 401

Considering negative numbers will practically makes \(N = ∞\) since you can consider integers up to \(-∞\)

Hence, Answer is A

I'm confused. The question doesn't specify prime factors, so wouldn't numbers with factors of 4, 5, 8, and 10 also be included?

The question as pointed out above should be worded better

To find the integers with non-common factors with 1000, let's first find the integers which do have factors in common with 1000.

To find that, there will be 3 types of integers, single digits, 2 digits and 3 digits.
for single digit numbers, the possible choices are - 0,2,4,6,8,5 hence 6C1.
for 2 digit numbers, 9C1*6C1
for 3 digit numbers, 9C1*10C1*6C1

Adding them all, 6+54+540=600

Hence the numbers which do not have common factors with 1000 = 1000-600 = 400
Hence, the answer should be C.

The question asks for the number of positive integers less than 1000 that have no factors (other than 1) in common with 1000. Since 1000 can be factored into \(2^3 \times 5^3\), we are essentially looking for integers that are not divisible by either 2 or 5.

Here's how we can break down the calculation:

1. **Multiples of 2**: The number of multiples of 2 in the range from 1 to 999 (since we are considering integers less than 1000) can be calculated by dividing 999 by 2 and rounding down to the nearest integer. This gives us the number of multiples of 2 as \(\left\lfloor \frac{999}{2} \right\rfloor\).

2. **Multiples of 5**: Similarly, the number of multiples of 5 in this range can be calculated by dividing 999 by 5 and rounding down. This gives us the number of multiples of 5 as \(\left\lfloor \frac{999}{5} \right\rfloor\).

3. **Multiples of both 2 and 5 (Multiples of 10)**: To avoid double counting the numbers that are multiples of both 2 and 5 (which are actually multiples of 10), we calculate this by dividing 999 by 10 and rounding down.

The total number of positive integers less than 1000 is 999. To find the number of integers that are not multiples of either 2 or 5, we subtract from 999 the sum of the multiples of 2 and 5, then add back the multiples of 10 (since they were subtracted twice, once in each set of multiples).

Let's calculate this properly:

- Total numbers: 999
- Multiples of 2: \(499\) (for \(2\) to \(998\))
- Multiples of 5: \(199\) (for \(5\) to \(995\))
- Multiples of both 2 and 5 (i.e., 10): \(99\) (for \(10\) to \(990\))

Therefore, the number of integers not divisible by either 2 or 5 is \(999 - (499 + 199 - 99)\).

The number of positive integers less than 1000 that have no factors (other than 1) in common with 1000 is \(400\). Therefore, the correct answer to the question is:

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