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

Question

If n is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does n have?

(A) four
(B) five
(C) six
(D) seven
(E) eight

EgmatQuantExpert · Accepted Answer

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. https://gmatclub.com/blog/wp-content/uploads/2015/05/Idea-Bulb-e1430745682443.png 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. https://gmatclub.com/blog/wp-content/uploads/2015/05/key-takeaway-stamp32-e1430746397281.png 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

Bunuel · Answer

SOLUTION

If n is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does n have?

(A) four
(B) five
(C) six
(D) seven
(E) eight

n=8! , so it has 4 prime factors: 2, 3, 5, and 7.

Answer: A.

BrushMyQuant · Answer

n = 1*2*3*4*5*6*7*8

so only Prime numbers which will be factors of n will be 2,3,5,7 (as prime numbers which are greater than 7 will not be there in the product of 1 to 8!)

So Answer is A (four)

Hope it Helps!

avohden · Answer

n = 8!

Prime factorization of 8!
 = 1, 2, 3, 2*2, 5, 2*3, 7, 2*2*2
 
 = 2, 3, 5, 7

= 4 prime factors > 1

JeffTargetTestPrep · Answer

n = 8 x 7 x 6 x 5 x 4 x 3 x 2

We can prime factorize and we have:

n = 2^7 x 3^2 x 5^1 x 7^1

Thus, n has 4 different prime factors.

Alternate solution:

In general, the number of distinct prime factors that k! (where k > 1) has is the number of prime numbers less than or equal to k. We have n = 8!, so k = 8; the number of prime numbers less than or equal to 8 is 4, namely, 2, 3, 5 and 7.

Answer: A

DaudDastagir · Answer

Bunuel 
Please explain 
instead of writing n=1x2x3x4x5x6x7x8
why did you write n=8!
What is the secrete?

Bunuel · Answer

The factorial of a  non-negative  integer  n , denoted by  n! , is the product of all  positive  integers less than or equal to  n .

For example:  4!=1*2*3*4=24 .

Check for more this:  Everything about Factorials on the GMAT

For other topics:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope it helps.

DanTheGMATMan · Answer

Just count the primes in the numbers 1-8. Watch for repeats::

https://www.youtube.com/watch?v=bjv1UD2Qi_4

GmatCracker09 · Answer

multiplication of 1 to 8 is divisible by 8! And 8! Has 2,3,5,7 as prime numbers.
So the Answer is (A).

Is my approach correct  KarishmaB

KarishmaB · Answer

Yes, that's all there is to this question!

gmatchile1 · Answer

Hello, why does the question indicate prime factor greater than 1? Aren't prime numbers all greater than 1?

KarishmaB · Answer

Yes by definition, prime numbers are natural numbers only i.e. positive integers. But sometimes, the notion of prime numbers is extended and applied to negative integers as well for certain academic queries. They are beyond our scope. If a question is clarifying, it is only to eliminate the possibililty of any kind of confusion or ambiguity.

bumpbot · Answer

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

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

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