Question

If M is the set of positive multiples of 2 less than 150 and N is the set of positive multiples of 9 less than 150, how many members are there in M \(\cap\) N?

• A. 0
• B. 8
• C. 9
• D. 18
• E. 74

Hint:

To find the number of multiples of a number 'n' up to a limit 'L', simply calculate L/n and take the integer part (floor) of the result. In this case, it was floor(149/18) or floor(150/18) since 150 is not a multiple.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question asks for the number of elements in the intersection of two sets, M and N. The intersection M \(\cap\) N contains elements that are common to both set M and set N.
Set M contains positive multiples of 2 less than 150.
Set N contains positive multiples of 9 less than 150.
Therefore, M \(\cap\) N contains numbers that are multiples of both 2 and 9, and are positive and less than 150.
Step 2: Key Formula or Approach:
A number that is a multiple of both 2 and 9 must be a multiple of their least common multiple (LCM).
\[ \text{LCM}(2, 9) = 18 \] So, we need to find the number of positive multiples of 18 that are less than 150.
Step 3: Detailed Explanation:
We need to find how many integers \(k\) exist such that \(18k<150\) and \(k>0\).
To find the number of multiples, we can divide 150 by 18.
\[ \frac{150}{18} \] We can simplify the fraction first:
\[ \frac{150 \div 6}{18 \div 6} = \frac{25}{3} \] \[ \frac{25}{3} = 8 \frac{1}{3} \approx 8.33 \] Since the number of multiples must be an integer, we take the integer part of this result. This means there are 8 multiples of 18 that are less than 150.
Let's list them to verify: 18, 36, 54, 72, 90, 108, 126, 144.
The next multiple would be \(144 + 18 = 162\), which is greater than 150.
There are indeed 8 such numbers.
Step 4: Final Answer:
There are 8 members in the set M \(\cap\) N.