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

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Math Expert
Joined: 02 Sep 2009
Posts: 58445
How many factors of 6! are greater than 100?  [#permalink]

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05 Nov 2017, 03:06
00:00

Difficulty:

65% (hard)

Question Stats:

52% (02:10) correct 48% (02:14) wrong based on 113 sessions

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How many factors of 6! are greater than 100?

(A) 4

(B) 6

(C) 7

(D) 10

(E) 12

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Math Expert
Joined: 02 Aug 2009
Posts: 7984
Re: How many factors of 6! are greater than 100?  [#permalink]

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05 Nov 2017, 05:30
1
Bunuel wrote:
How many factors of 6! are greater than 100?

(A) 4

(B) 5

(C) 7

(D) 10

(E) 12

A good Q and not a very easy one ...

$$6! =1*2*3*4*5*6 = 2^4*3^2*5=720$$
now the trick is NOT to try and find the ONES greater than 100 BUT the ones that are required to be multiplied by factors above 100 to get 6!
the lowest factors are 1,2,3,4,5,6 which can be seen from 6! if we check 6... $$\frac{6!}{6}=2*3*4*5=120>100$$
next will be 2*4=8 and $$\frac{2*3*4*5*6}{2*4}=3*5*6=90<100$$
so our answer is 1,2,3,4,5,6 so 6 values

Realized, missed out on 1....
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Re: How many factors of 6! are greater than 100?  [#permalink]

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05 Nov 2017, 07:03
1
1
6!= 6*5*4*3*2
What combination of numbers is over 100?
1. 6*5*4
2. 6*5*4*3
3. 6*5*4*3*2
4. 6*5*4*2
5. 6*5*3*2

Note: 5*4*3*2 is over 100, but it can be written as (3*2)*5*4, which is equal 6*5*4

Ans: B

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Re: How many factors of 6! are greater than 100?  [#permalink]

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05 Nov 2017, 07:38
1
Bunuel wrote:
How many factors of 6! are greater than 100?

(A) 4

(B) 5

(C) 7

(D) 10

(E) 12

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

Now number of factors that are greater than 100 are -

1) 6*5*4*3*2 = 720

2) 6*5*4*3 = 360

3) 6*5*4*2 = 240

4) 6*5*3*2 = 180

5) 6*4*3*2 = 144

6) 5*4*3*2 = 120

IMO number of factors should be 6 but there is no such option

Hi Bunuel

Can you confirm whether the OAs are correct or point out the EXTRA factor in my method?
Math Expert
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Posts: 58445
Re: How many factors of 6! are greater than 100?  [#permalink]

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05 Nov 2017, 09:35
chetan2u wrote:
Bunuel wrote:
How many factors of 6! are greater than 100?

(A) 4

(B) 5

(C) 7

(D) 10

(E) 12

A good Q and not a very easy one ...

$$6! =1*2*3*4*5*6 = 2^4*3^2*5=720$$
now the trick is NOT to try and find the ONES greater than 100 BUT the ones that are required to be multiplied by factors above 100 to get 6!
the lowest factors are 1,2,3,4,5,6 which can be seen from 6! if we check 6... $$\frac{6!}{6}=2*3*4*5=120>100$$
next will be 2*4=8 and $$\frac{2*3*4*5*6}{2*4}=3*5*6=90<100$$
so our answer is 1,2,3,4,5,6 so 6 values

Realized, missed out on 1....

The OA is 6. Edited the options. Thank you.
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Re: How many factors of 6! are greater than 100?  [#permalink]

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04 Jan 2018, 16:57
1
Hi All,

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).

B; 6 factors

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Re: How many factors of 6! are greater than 100?  [#permalink]

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09 May 2019, 09:48
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Re: How many factors of 6! are greater than 100?   [#permalink] 09 May 2019, 09:48
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