If W is the set of all the integers between 49 and 76, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?  Options
A) 19 B) 91 C) 41 D) 18 E) 14
Official Solution:   Number of multiples of 3   Step 1. Subtract the extreme multiples of 3 within the range (The greatest is 75, the smallest is 51): 75 - 51 = 24   Step 2. Divide by 3: 24 / 3 = 8   Step 3. Add 1: 8 + 1 = 9.   So there are 9 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 76, the smallest is 50): 76 - 50 = 26   Step 2. Divide by 2: 26 / 2 = 13   Step 3. Add 1: 13 + 1 = 14.   So there are 14 multiples of 2 within the range: examples are 50, 52, 54, 56, 58, 60 etc.   Add the 9 multiples of 3 and the 14 multiples of 2: 9+14=23.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   9+14= 
<gadget id="calculator">  9+14</gadget>
<output>23</output>
23. 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 23:   Step 1. Subtract the extreme multiples of 6 within the range (The greatest is 72, the smallest is 54): 72 - 54= 18   Step 2. Divide by 6: 18 / 6 = 3   Step 3. Add 1: 3 + 1 = 4.   So there are 4 multiples of 6 within the range: we counted 4 numbers twice.   Subtract the 4 multiples of 6 from the sum of the multiples of 2 and 3:   = 9 + 14 - 4   = 23 - 4= 
<gadget id="calculator">23 - 4</gadget>
<output>19</output>
19   Therefore, the final number of multiples of 2, 3 or 6 is 19.   Hence, this is the correct answer. (19)
<result>A</result>
If W is the set of all the integers between 49 and 76, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?