If n is the product of the integers from 1 to 8, inclusive

In this question, I have noticed that many students are prime factorizing every term in the product to find out the answer. But is that necessary?

What if the expression was Z = 1*2*3*…*30. Would you have factorized every term?

Let's do a quick concept recap.

Concept Recap: Primes are the basic building blocks for every positive integer greater than 1. Every positive integer greater than 1 is itself a prime or a product of primes less than the number itself.

How is this related to the question?: Take the example of 6!. 6! as we all know is equal to \(1*2*3*4*5*6\). Obviously, we don't need to factorize every element in this expression to find out the different prime factors of 6!. Using the knowledge from the above concept recap that the different prime factors of 6! will be simply the prime numbers less than or equal to 6 itself, we can say the prime factors of 6! are 2, 3 and 5. Therefore 6! has 3 prime factors.

Answer for this question: Primes less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 (a total of 10 primes.). Therefore, if Z = 30!, then there would 10 different prime factors for Z.

In such questions, where you need to find the number of prime factors of a factorial expression, do not waste your time factorizing every term. Number of prime factors of n! will be simply the number of prime numbers less than n.



Footnote for the curious minded: It would have made sense to factorize every term in the expression, if the question had asked the "total number of factors" instead of "number of prime factors". To find the total number of factors, we definitely would need to find the prime factors and their powers in the expression.

You can take a stab at the following questions to test your understanding of these concepts.

x-is-the-largest-prime-number-less-than-positive-integer-n-p-is-an-in-197329.html

p-is-the-smallest-perfect-cube-greater-than-197336.html


Regards,
Krishna