If W is the set of all the integers between 49 and 114, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?
Pick
A) 34
B) 44
C) 45
D) 55
E) 72
Official Solution:   Number of multiples of 3   Step 1. Subtract the extreme multiples of 3 within the range (The greatest is 114, the smallest is 51): 114 - 51 = 63   Step 2. Divide by 3: 63 / 3 = 21   Step 3. Add 1: 21 + 1 = 22.   So there are 22 multiples of 3 within the range: examples are 51, 54, 57, 60, etc.   Number of multiples of 2   Step 1. Subtract the extreme multiples of 2 within the range (The greatest is 114, the smallest is 50): 114 - 50 = 64   Step 2. Divide by 2: 64 / 2 = 32   Step 3. Add 1: 32 + 1 = 33.   So there are 33 multiples of 2 within the range: examples are 50, 52, 54, 56, 58, 60 etc.   Add the 22 multiples of 3 and the 33 multiples of 2: 22+33=55.However, by adding the multiples of 2 and the multiples of 3, we are effectively counting several numbers twice: for example, 54 and 60 are parts of both the lists above. So we can't just take   22+33= 
<gadget id="calculator">  22+33</gadget>
<output>55</output>
55. Find the Number of multiples of 6 (which are the double counted, as 6 is divisible by both 2 and 3), and subtract it from 55:   Step 1. Subtract the extreme multiples of 6 within the range (The greatest is 72, the smallest is 54): 114 - 54= 60   Step 2. Divide by 6: 60 / 6 = 10   Step 3. Add 1: 10 + 1 = 11.   So there are 11 multiples of 6 within the range: we counted 11 numbers twice.   Subtract the 11 multiples of 6 from the sum of the multiples of 2 and 3:   = 22+33 - 11   = 55 - 11= 
<gadget id="calculator">55 - 11</gadget>
<output>44</output>
44   Therefore, the final number of multiples of 2, 3 or 6 is 44.   Hence, this is the correct answer. (44)
<result>B</result>
If W is the set of all the integers between 49 and 114, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?