Question

The set S={1,2,3,…,49,50} contains the first 50 positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set S?

a. 16
b. 17
c. 25
d. 33

Answer 1

The correct answer is option d. 33. To find the integers left in the set after removing multiples of 2 and 3, the principle of inclusion-exclusion is used.

Step-by-step explanation:

To find the remaining integers in the set S after removing the multiples of 2 and the multiples of 3, we can use the principle of inclusion-exclusion. First, we calculate the number of multiples of 2 within the first 50 positive integers:

• 50 ÷ 2 = 25 multiples of 2

Then, we find the number of multiples of 3 within the same range:

• 50 ÷ 3 = 16 multiples of 3

However, some numbers are multiples of both 2 and 3 (in other words, multiples of 6). We need to find these common multiples so we don't subtract them twice:

• 50 ÷ 6 = 8 multiples of 6

Now we subtract the multiples of 2 and 3 from the total count of 50, and add back the common multiples:

• 50 - (25 + 16 - 8) = 50 - 33 = 17

However, this answer does not include number 1, which is neither a multiple of 2 nor 3, so we need to add 1 to our result:

• 17 + 1 = 18

Therefore, after removing the multiples of 2 and 3, 17 integers remain in the set S.