if w is the set of all the integers between 49 and 88 , inclusive , that are either multiples of 3 or multiples of 2 or multiples of both , then w contains how many numbers ?
"official solution : number of multiples of 3 step 1 . subtract the extreme multiples of 3 within the range ( the greatest is 87 , the smallest is 51 ) : 87 - 51 = 36 step 2 . divide by 3 : 36 / 3 = 12 step 3 . add 1 : 12 + 1 = 13 . so there are 13 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 88 , the smallest is 50 ) : 88 - 50 = 38 step 2 . divide by 2 : 38 / 2 = 19 step 3 . add 1 : 19 + 1 = 20 . so there are 19 multiples of 2 within the range : examples are 50 , 52 , 54 , 56 , 58 , 60 etc . add the 13 multiples of 3 and the 19 multiples of 2 : 13 + 19 = 32 . 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 ca n ' t just take 13 + 19 = 32 . 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 32 : step 1 . subtract the extreme multiples of 6 within the range ( the greatest is 72 , the smallest is 54 ) : 84 - 54 = 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 : = 13 + 19 - 6 = 32 - 6 = 26 therefore , the final number of multiples of 2 , 3 or 6 is 26 . hence , this is the correct answer . ( e )"
subtract(add(floor(divide(subtract(88, 49), 3)), divide(subtract(88, 49), 2)), floor(divide(subtract(88, 49), multiply(2, 3))))