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## How many factors of 6! are greater than 100?

This topic has 2 expert replies and 0 member replies

### Top Member

ardz24 Master | Next Rank: 500 Posts
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#### How many factors of 6! are greater than 100?

Sun Nov 05, 2017 4:19 am
How many factors of 6! are greater than 100?

(A) 4
(B) 5
(C) 7
(D) 10
(E) 12

I'm confused how to set up the formulas here. Can any experts help?

### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
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GMAT Score:
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Sun Nov 05, 2017 12:29 pm
Hi ardz24,

We're asked for the number of factors of 6! that are greater than 100. Since 6! is a relatively easy number to deal with, we can 'brute force' the solution by listing out the factors that are greater than 100:
(1)(720)
(2)(360)
(3)(240)
(4)(180)
(5)(144)
(6)(120)
(8)(90)

At this point, we can stop working. Each of the first 6 pairs has a factor greater than 100, but as the first number in each pair increases, the second number will decrease (thus, there's no reason to look at 'pairs' above 8 and 90).

GMAT assassins aren't born, they're made,
Rich

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### GMAT/MBA Expert

GMATGuruNY GMAT Instructor
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Sun Nov 05, 2017 5:37 am
ardz24 wrote:
How many factors of 6! are greater than 100?

(A) 4
(B) 5
(C) 7
(D) 10
(E) 12
Start with the GREATEST FACTOR and work down to the smallest factor greater than 100.

6! = 6*5*4*3*2.

Case 1: Combine 5 of the values in blue to yield a product greater than 100
6*5*4*3*2
Total options = 1.

Case 2: Combine 4 of the values in blue to yield a product greater than 100
6*5*4*3
6*5*4*2
6*5*3*2
6*4*3*2
5*4*3*2
Total options = 5.

Case 3: Combined 3 of the values in blue to yield a product greater than 100
6*5*4
Discard this option, since 6*5*4 is the same as 5*4*3*2, counted above.

Total options = Case 1 + Case 2 = 1+5 = 6.

The OA is not among the answer choices.
What is the source of this problem?

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