In set of 30 different positive integers, every number is even and/or a multiple of 3. If 22 of these numbers are even and 15 are multiple of 3, then how many of these numbers are multiple of 6?

all multiples of 6 are even and multiples of 3

Venn diagram (intersection is mult of 6)
n are multiples of 6
22 - n are even only
15 - n are 3 only

(22- n) + n + (15-n) = 30

37 - n = 30

n = 7

To determine how many of these numbers are multiples of 6, we need to consider the common factors of even numbers and multiples of 3.

An even number must be divisible by 2, which means it has a factor of 2. A multiple of 3 must be divisible by 3, so it has a factor of 3.

To find the count of numbers that are multiples of 6, we need to find the numbers that have both factors of 2 and 3.

Step 1: Count the numbers that are even (factor of 2): 22
Step 2: Count the numbers that are multiples of 3 (factor of 3): 15

Now, to find the numbers that are multiples of 6 (factors of both 2 and 3), we need to find the intersection of these two sets.

Step 3: Determine the common multiples of 2 and 3, which is the least common multiple (LCM) of 2 and 3: LCM(2, 3) = 6

Therefore, the count of numbers that are multiples of 6 is the same as the count of numbers that have both factors of 2 and 3.

Step 4: Find the count of numbers that are multiples of 6: the minimum count of even numbers and multiples of 3

In this case, the count of numbers that are multiples of 6 is 15 (since there are 15 multiples of 3).

So, in the set of 30 different positive integers, there are 15 numbers that are multiples of 6.