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## OG If n is the product

tagged by: Brent@GMATPrepNow

This topic has 5 expert replies and 0 member replies
AbeNeedsAnswers Master | Next Rank: 500 Posts
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#### OG If n is the product

Tue Jul 25, 2017 8:16 pm
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

A

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Scott@TargetTestPrep GMAT Instructor
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Mon Aug 14, 2017 11:52 am
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

A
n = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

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.

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Matt@VeritasPrep GMAT Instructor
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Fri Aug 18, 2017 2:15 pm
Count the unique primes in 2 * 3 * 4 * 5 * 6 * 7 * 8

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### GMAT/MBA Expert

Scott@TargetTestPrep GMAT Instructor
Joined
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Mon Aug 14, 2017 11:52 am
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

A
n = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

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.

_________________

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GMAT Quant Self-Study Course - 500+ lessons 3000+ practice problems 800+ HD solutions
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### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
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Fri Aug 18, 2017 2:15 pm
Count the unique primes in 2 * 3 * 4 * 5 * 6 * 7 * 8

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Rich.C@EMPOWERgmat.com Elite Legendary Member
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Wed Jul 26, 2017 6:52 pm

With these types of Prime Factorization questions, it often helps to 'break down' the math into 'pieces' (since the individual pieces are rarely all that difficult to deal with.

Here, we're asked for the number of different prime factors in the product of 1 to 8, inclusive (essentially 8!). That product would include the following factors:
1
2
3
4 = (2)(2)
5
6 = (2)(3)
7
8 = (2)(2)(2)

Thus, the different primes are: 2, 3, 5 and 7... and there are 4 of them.

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Rich

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