If W is the set of all the integers between 35 and 64, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?
Options:
A) 18	B) 19	C) 61	D) 31	E) 11
Official Solution:   Number of multiples of 3   Step 1. Subtract the extreme multiples of 3 within the range (The greatest is 63, the smallest is 36): 63 - 36 = 27   Step 2. Divide by 3: 27 / 3 = 9   Step 3. Add 1: 9 + 1 = 10.   So there are 10 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 64, the smallest is 36): 64 - 36 = 28   Step 2. Divide by 2: 28 / 2 = 14   Step 3. Add 1: 14 + 1 = 15.   So there are 15 multiples of 2 within the range: examples are 50, 52, 54, 56, 58, 60 etc.   Add the 10 multiples of 3 and the 15 multiples of 2: 10+15=25.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   10+15= 
<gadget id="calculator">  10+15</gadget>
<output>25</output>
25. 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 25:   Step 1. Subtract the extreme multiples of 6 within the range (The greatest is 72, the smallest is 54): 60 - 30= 30   Step 2. Divide by 6: 30 / 6 = 5   Step 3. Add 1: 5 + 1 = 6.   So there are 6 multiples of 6 within the range: we counted 6 numbers twice.   Subtract the 6 multiples of 6 from the sum of the multiples of 2 and 3:   = 10+15 - 6   = 25 - 6= 
<gadget id="calculator">25 - 6</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>B</result>
If W is the set of all the integers between 35 and 64, inclusive, that are either multiples of 3 or multiples of 2 or multiples of both, then W contains how many numbers?