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How many factors of 6! are greater than 100?

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by GMATGuruNY » 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.
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by [email protected] » 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).

Final Answer: 6 factors

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by Scott@TargetTestPrep » Tue Oct 29, 2019 6:01 am
GMATGuruNY wrote:
ardz24 wrote:How many factors of 6! are greater than 100?

(A) 4
(B) 5
(C) 7
(D) 10
(E) 12
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

Now let's write the factors from largest to smallest, with their cofactor in parentheses in the row below:

720, 360, 240, 180, 144, 120, 90, ...
(1), (2), (3), (4), (5), (6), (8), ...

We see that there are 6 factors greater than 100.

Answer: 6

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