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Question

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?

A. 23
B. 24
C. 25
D. 26
E. 27

Answer

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.

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