Here is a new Challenge Problem! If you want to win prizes, try entering our Challenge Problem Showdown. The more people that enter our challenge, the better the prizes!
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#42?
First, translate the expression 21#42, using the definition given:
21#42 = 21×24×27×30×33×36×39×42
You need the prime factorization of this enormous product.
Since these are consecutive multiples of 3, a good move is to factor out that 3 from each multiple. You have 8 multiples, all multiplied together, so you get :
21#42 = (7×8×9×10×11×12×13×14)
Now replace each consecutive integer with its prime factorization:
21#42 = (7×××(2×5)×11×(×3)×13×(2×7))
Group up the prime bases:
21#42 = ××5××11×13
Don’t forget to put in an understood 1 for the primes lacking explicit exponents:
21#42 = ×××××
Finally, add up the exponents:
7 + 11 + 1 + 2 + 1 + 1 = 23
The correct answer is A.
Special Announcement: If you want to win prizes for answering our Challenge Problems, try entering our Challenge Problem Showdown. Each week, we draw a winner from all the correct answers. The winner receives a number of our our Strategy Guides. The more people enter, the better the prize. Provided the winner gives consent, we will post his or her name on our Facebook page.