Question

How many numbers are there between 1 and 1000(both included) that are not
divisible by 2, 3, and 5?
​

Answer 1

Answer:

All the answers mentioned are right. Yes, this can be solved using a Venn diagram and intersections.

But, if you are going to solve it in an exam, I would suggest a better and faster approach.

Here, in our case LCM of 2,3 and 5 is 30.

So, we can say that, if we write from 1 to 30 and find out the pattern, the same would be repeated across sets of 30 numbers thereafter.

Step 1

Write all numbers from 1 to 30

Step 2

Start with 2, the smallest divisor and cross out the numbers that are divisible. Then move to the next smallest one, in this case 3, circle out the numbers divisible by 3. And then do it for 5, use squares. The sheet would look like this.

Step 3

Now that you need the numbers that are not divisible by either of these factors, let’s list them out.

Step 4

Finding the answer

1000 = 30 * 33 + 10

So, the numbers not divisible by 2,3 or 5 would be 8 * 33 + 2 = 264 + 2 = 266

Note:

This method can also be used to find the number of numbers between ranges that are divisible by specific factors. We need to start the count from the smallest number in the range and then list down a series of consecutive numbers (total count = LCM of factors)

This method can also be used to find the number of numbers divisible by 2,3 and 5. In that case, instead of listing down the left ones, list down the crossed / highlighted ones.

This method works faster when you have to do it on a small subset and extend the results. If the LCM is more than 100 or the factors are large numbers, Venn diagram method would be the way to go ahead.

Answer 2

$\large{\boxed{\boxed{\sf{Question}}}}$

How many numbers are there between 1 and 1000(both included) that are not divisible by 2, 3, and 5?

$\large{\boxed{\boxed{\sf{Answer}}}}$

= 266

$\large{\boxed{\boxed{\sf{Let}}}}$

• A denote the set of numbers that are divisible by 2

• B set of numbers that are

divisible by 3

• C set of numbers that are divisible by 5

• D set of numbers that are divisible by

both 2 and 3

• E set of numbers that are divisible by both 2 and 5

• F set of numbers that are

divisible by 3 and 5

• G set of numbers that are divisible by all the three numbers

$\large{\boxed{\boxed{\sf{Solution}}}}$

$\sf a + (n - 1)d = T_n$

$\sf n = \dfrac{T_n}{d} - \dfrac{a}{d} + 1$

In this case

$\sf \dfrac{a}{d} = 1$ , Hence n = Integer part of $\sf \dfrac{T_n}{d}$

$\sf n(A) = [\dfrac{1000}{2}] = 500$

$\sf n(B) = [\dfrac{1000}{3}] = 333$

$\sf n(C) = [\dfrac{1000}{5}] = 200$

$\sf n(D) = [\dfrac{1000}{2 \times 3}] = 166$

$\sf n(E) = [\dfrac{1000}{2 \times 5}] = 100$

$\sf n(F) = [\dfrac{1000}{3 \times 5}] = 66$

Number that are divisible by 2 , 3 ,5 are

n ( A ∪ B ∪ C )

= n ( A ) + n ( B ) + n ( C ) - n ( A ∪ B ) - n ( A ∪ C ) - n ( B ∪ C ) - n ( A ∩ B ∩ C )

= 500 + 333 + 200 + 166 + 100 + 66 + 33

= 734

Number that are not divisible by 2 , 3 , 5 are

= 1000 - 734 = 266