the expression x # y denotes the product of the consecutive multiples of 3 between x and y , inclusive . what is the sum of the exponents in the prime factorization of 21 # 36 ?
first , let ' s translate the expression 21 # 36 , using the definition given : 21 # 36 = 21 × 24 × 27 × 30 × 33 × 36 we need the prime factorization of this product . let ' s factor out 3 from each multiple . 21 # 36 = 3 ^ 6 ( 7 × 8 × 9 × 10 × 11 × 12 ) now let ' s replace each consecutive integer with its prime factorization : 21 # 36 = 3 ^ 6 ( 7 × 2 ^ 3 × 3 ^ 2 × ( 2 × 5 ) × 11 × ( 2 ^ 2 × 3 ) ) let ' s group the prime bases : 21 # 36 = 2 ^ 6 × 3 ^ 9 × 5 × 7 × 11 the sum of the exponents is 6 + 9 + 1 + 1 + 1 = 18 the answer is c .
add(add(add(3, add(const_2, const_2)), add(const_2, const_2)), add(3, add(const_2, const_2)))