Question

$x$ is number of numbers between 100 and 200 such that $x$ is odd and $x$ is divisible by 3 but not by 7. What is $x$?

• A. 16
• B. 12
• C. 11
• D. 13

Hint:

Count under multiple restrictions using inclusion-exclusion: count first condition, subtract those violating second condition.

Answer 1

The Correct Option is D

Step 1: Find odd multiples of 3 in the range

The smallest multiple of 3 greater than 100 that is odd is \(105\). 
The largest multiple of 3 less than 200 that is odd is \(195\). 
Therefore, the sequence is: \(105, 111, 117, \dots, 195\) with a common difference of \(6\).

Number of terms: \[ n = \frac{195 - 105}{6} + 1 = \frac{90}{6} + 1 = 15 + 1 = 16 \]

Step 2: Remove those divisible by 7

The LCM of 3 and 7 is \(21\). 
The odd multiples of 21 between 100 and 200 are: \(105, 147, 189\). 
Count = \(3\).

Step 3: Subtract

\[ 16 - 3 = 13 \]

Final Answer:

\[ \boxed{x = 13} \]