Hi
need some help with the following
Question
Which of the following is an integer
1. 12!/6!
2. 12!/8!
3. 12!/7!5!
esp with the 3.12!/7!5!
Thanks
Factorials & divisibility
This topic has expert replies
- DavidG@VeritasPrep
- Legendary Member
- Posts: 2663
- Joined: Wed Jan 14, 2015 8:25 am
- Location: Boston, MA
- Thanked: 1153 times
- Followed by:128 members
- GMAT Score:770
Let's write these expressions out. If we can completely cancel out the denominator, then we'll be left with an integer value.
1) (12*11*10*9*8*7*6*5*4*3*2*1)/(6*5*4*3*2*1)
We can cancel out a 6! in both numerator and denominator, which leaves us with 12*11*10*9*8*7. No need to calculate. Clearly an integer.
2) (12*11*10*9*8*7*6*5*4*3*2*1)/(8*7*6*5*4*3*2*1)
We can cancel out an 8! in both numerator and denominator, which leaves us with 12*11*10*9. Again, no need to calculate. Clearly an integer.
3)(12*11*10*9*8*7*6*5*4*3*2*1)/[(7*6*5*4*3*2*1)*(5*4*3*2*1)]
We can cancel out a 7! in both numerator and denominator, which leaves us with (12*11*10*9*8)/(5*4*3*2*1)
The 5 and the 2 in the denominator will cancel out the 10 in the numerator
Now we have (12*11*9*8)/(4*3*1)
The 4 and the 3 in the denominator will cancel out the 12 in the numerator
We're left with 11*9*8. Clearly an integer.
1) (12*11*10*9*8*7*6*5*4*3*2*1)/(6*5*4*3*2*1)
We can cancel out a 6! in both numerator and denominator, which leaves us with 12*11*10*9*8*7. No need to calculate. Clearly an integer.
2) (12*11*10*9*8*7*6*5*4*3*2*1)/(8*7*6*5*4*3*2*1)
We can cancel out an 8! in both numerator and denominator, which leaves us with 12*11*10*9. Again, no need to calculate. Clearly an integer.
3)(12*11*10*9*8*7*6*5*4*3*2*1)/[(7*6*5*4*3*2*1)*(5*4*3*2*1)]
We can cancel out a 7! in both numerator and denominator, which leaves us with (12*11*10*9*8)/(5*4*3*2*1)
The 5 and the 2 in the denominator will cancel out the 10 in the numerator
Now we have (12*11*9*8)/(4*3*1)
The 4 and the 3 in the denominator will cancel out the 12 in the numerator
We're left with 11*9*8. Clearly an integer.
GMAT/MBA Expert
- [email protected]
- Elite Legendary Member
- Posts: 10392
- Joined: Sun Jun 23, 2013 6:38 pm
- Location: Palo Alto, CA
- Thanked: 2867 times
- Followed by:511 members
- GMAT Score:800
Hi binaras,
Beyond the arithmetic that David explained, these calculations fall into some patterns that you are likely to see (in some form) on Test Day.
12!/6!
When dealing with INDIVIDUAL factorials, if the factorial in the numerator is GREATER than the factorial in the denominator, then you will end up with an integer. Here, since 12 > 6, 12!/6! will simplify to an integer.
12!/7!5!
You might not have studied the Combination Formula yet, but this calculation is what you would end up with when answering the question "how many different combinations of 7 people can you form from a group of 12 people?" In these types of "Combination" calculations, the number of groups will always be an integer, so 12!/7!5! will simplify to an integer.
GMAT assassins aren't born, they're made,
Rich
Beyond the arithmetic that David explained, these calculations fall into some patterns that you are likely to see (in some form) on Test Day.
12!/6!
When dealing with INDIVIDUAL factorials, if the factorial in the numerator is GREATER than the factorial in the denominator, then you will end up with an integer. Here, since 12 > 6, 12!/6! will simplify to an integer.
12!/7!5!
You might not have studied the Combination Formula yet, but this calculation is what you would end up with when answering the question "how many different combinations of 7 people can you form from a group of 12 people?" In these types of "Combination" calculations, the number of groups will always be an integer, so 12!/7!5! will simplify to an integer.
GMAT assassins aren't born, they're made,
Rich
GMAT/MBA Expert
- Jeff@TargetTestPrep
- GMAT Instructor
- Posts: 1462
- Joined: Thu Apr 09, 2015 9:34 am
- Location: New York, NY
- Thanked: 39 times
- Followed by:22 members
Solution:binaras wrote:Hi
need some help with the following
Question
Which of the following is an integer
1. 12!/6!
2. 12!/8!
3. 12!/7!5!
esp with the 3.12!/7!5!
Thanks
Before actually solving this problem, let's review how factorials can be expanded and expressed. As as example, we can use 5!.
5! could be expressed as:
5!
5 x 4!
5 x 4 x 3!
5 x 4 x 3 x 2!
5 x 4 x 3 x 2 x 1!
Understanding how this factorial expansion works will help us work our way through each answer choice, especially answer choices 1 and 2.
1. 12!/6!
Since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8 x 7 x 6!
Plugging this in for answer choice 1, we have:
(12 x 11 x 10 x 9 x 8 x 7 x 6!)/6! = 12 x 11 x 10 x 9 x 8 x 7, which is an integer.
2. 12!/8!
Once again, since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8!
Plugging this in for answer choice 2, we have:
(12 x 11 x 10 x 9 x 8!)/8! = 12 x 11 x 10 x 9, which is an integer.
3. 12!/(7!5!)
Once again, since we know that factorials can be expanded, we now know that:
12! = 12 x 11 x 10 x 9 x 8 x 7!
Plugging this in for answer choice 3 gives us:
(12 x 11 x 10 x 9 x 8 x 7!)/(7!5!)
(12 x 11 x 10 x 9 x 8)/(5 x 4 x 3 x 2 x 1)
(12 x 11 x 10 x 9 x 8)/(12 x 10 x 1)
11 x 9 x 8, which is an integer.
We see that choices 1, 2, and 3 are all integers.
Jeffrey Miller
Head of GMAT Instruction
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews