How many integers from 1 through 1000 are neither multiple o

How many integers from 1 through 1000 are neither multiple of 2 nor multiple of 9?

Solution

We can go this way:

Calculate the no. of terms from 1 to 1000 (inclusive) that are divisible by 2 or 9 or both.

1.) Total no. of terms divisible by 2 are 500. We can calculate this by finding the first and last terms, which are 2 & 1000 respectively. Then we will find the total no. of terms by using equation

Last Term = a + (n-1)d where a=2, d=2, Last Term=1000.

So, n=500

2.) Similarly, total no. of terms divisible by 9 are 111. Find it using the above method.

3.) To find terms divisible by both 2 & 9, find the first term. Since both have no common factors except 1, just multiply 2 & 9 to get the first common term i.e., 18. Next term is 36.

So, in total, there are 55 common terms for 2 & 9 both.
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Hence, the total no. of terms from 1 to 1000 (inclusive) that are divisible by 2 or 9 or both = 500 + 111 - 55 = 556

So, the correct answer = 1000 - 556 = 444, which will give us the total no. of terms that are divisible neither by 2 nor 9.