Question

The number of all even integers between 99 and 999 which are not multiples of 3 and 5 is

• A. 240
• B. 250
• C. 235
• D. 245

Hint:

Use the inclusion-exclusion principle to count the number of integers divisible by 3 or 5, and subtract from the total number of even integers.

Answer 1

The Correct Option is A

Step 1: The even integers between 99 and 999 are all integers of the form \( 2k \) where \( 100 \leq 2k \leq 998 \). Thus, the first even number is 100, and the last is 998. The total number of even integers is: \[ \frac{998 - 100}{2} + 1 = 450 \] Step 2: Next, we need to exclude those that are divisible by 3 or 5. - The number of even integers divisible by 3 is: \[ \frac{998 - 102}{6} + 1 = 150 \] - The number of even integers divisible by 5 is: \[ \frac{1000 - 100}{10} = 90 \] - The number of even integers divisible by both 3 and 5 (i.e., divisible by 15) is: \[ \frac{990 - 120}{30} + 1 = 30 \] Step 3: Using the inclusion-exclusion principle, the number of even integers divisible by 3 or 5 is: \[ 150 + 90 - 30 = 210 \] Step 4: The number of even integers that are not divisible by 3 or 5 is: \[ 450 - 210 = 240 \]