Find The Non-multiples

Let $M_2$ be the set of integers between $1$ and $1000$ (inclusive) that are multiples of $2$ .

Let $M_5$ be the set of integers between $1$ and $1000$ (inclusive) that are multiples of $5$ .

The question is asking for $\left|M_2^c\cap M_5^c\right|$ . By De Morgan's Laws , it suffices to find $\left|\left(M_2\cup M_5\right)^c\right|$ instead.

$|M_2|=500$

$|M_5|=200$

$|M_2\cap M_5|=100$

By PIE , $|M_2\cup M_5|=500+200-100=600$ .

Thus, $\left|\left(M_2\cup M_5\right)^c\right|=1000-600=400$ .

The number of integers between $1$ and $1000$ that are neither multiples of $2$ nor multiples of $5$ is $\boxed{400}$ .

There are:

• 1000 ÷ 2 = 500 integers between 1 and 1000 (inclusive), which are divisible by 2;

• 1000 ÷ 5 = 200 integers between 1 and 1000 (inclusive), which are divisible by 5; and

• 1000 ÷ (2 × 5) = 100 (double counted) integers between 1 and 1000 (inclusive), which are divisible by both 2 and 5 .

Now, we can find the number of integers between 1 and 1000 (inclusive), which are both multiples of 2 and multiples of 5 by applying the principles of inclusion - exclusion:

500 + 200 - 100 = 600

Hence, the number of integers between 1 and 1000 (inclusive), which are both multiples of 2 and multiples of 5:

$1000 - 600 = \boxed {400}$

• Lets Narrow $1000$ to $10$ .
• Eliminate even numbers as they can be divided by $2$ . The numbers left are :1,3,5,7,9
• Eliminate the numbers that can be divided by $5$ , including itself. The numbers left are :1,3,7,9
• Since there are $4$ out of $10$ numbers that are neither multiples of $2$ and $5$ , the fraction $\frac{4}{10}$ comes out as a formula for the question.
• $\frac{4 }{10}$ of $1000$ will be $400$ .

As $2$ and $5$ are both factors of $1000$ , we can work out how many integers between $1$ and $lcm(2,5)$ (inclusive) are neither multiples of $2$ nor multiples of $5$ , and then multiply this result by $\frac{1000}{lcm(2,5)}$ .

$lcm(2,5) = 10$ . The integers between $1$ and $10$ (inclusive) not divisible by $2$ or $5$ are $1$ , $3$ , $7$ , and $9$ . This means there are $4$ integers in this range not divisible by $2$ or $5$ .

We now need to multiply this amount by $\frac{1000}{lcm(2,5)} = \frac{1000}{10} = 100$ . This is equal to $4*100 = 400$ .

1000 (1-1/2) (1-1/5) = 400

count the numbers that arent divisable by 2 or 5 that are between 1 and 10 (you get 4) multiple by 10 to get how many are between 1 and 100 then multiply by 10 again to get between 1 and 1000. Quit complicating things guys!

Integer multiples of 2 within range of 1 to 1000 is half of the number, which is 500. Within the 500 odd numbers remaining, every 5 of them is a multiple of 5, therefore 100 of those numbers are mulitples of 5. This leaves 400 that are odd numbers and not multiples of 5.