by: DeepGOO225
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by: DeepGOO225
15 mins
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Difficulty:
95%
(hard)
Question Stats:
49% (02:36) correct
51%
(02:41)
wrong
based on 134
sessions
History
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How many numbers are there between 1 and 30 including both the numbers such that n! is divisible by (n + 1)?
A. 17
B. 20
C. 18
D. 30
E. 19
A. 17
B. 20
C. 18
D. 30
E. 19
Posts: 34
We can solve by the following method:-
1 can be written as 1!/1+1 i.e 1/2 which is not properly divisible
2 can be written as 2*1/3 which is also not properly divisible
3 can be written as 3*2*1/4 or 3*2*1/2*2 which is also not properly divisible
4 cab be written as 4*3*2*1/5 which is also not properly divisible.
From this we can say that if n+1 is a prime no., we can eliminate that no. from the possible no. which can divide n! by n+1. So the no. we can eliminate are 1,2,4,6,10,12,16,18,22,28 & 30. So we already know 3 can also be eliminated from the no. that can divide n! by n+1. So total 12 numbers are there that cannot divide n! by n+1 properly
So our answer will be 18 no. between 1 to 30 that n! is divisible by n+1
1 can be written as 1!/1+1 i.e 1/2 which is not properly divisible
2 can be written as 2*1/3 which is also not properly divisible
3 can be written as 3*2*1/4 or 3*2*1/2*2 which is also not properly divisible
4 cab be written as 4*3*2*1/5 which is also not properly divisible.
From this we can say that if n+1 is a prime no., we can eliminate that no. from the possible no. which can divide n! by n+1. So the no. we can eliminate are 1,2,4,6,10,12,16,18,22,28 & 30. So we already know 3 can also be eliminated from the no. that can divide n! by n+1. So total 12 numbers are there that cannot divide n! by n+1 properly
So our answer will be 18 no. between 1 to 30 that n! is divisible by n+1
why 4 is there because 3factorial is not divided by 4
From 1! to 4! nothing is divisble. from 5! if we observe every no is divisble except from prime no, from 5 to 30 we have 25 no. minus prime no. between 6 to 30 ie 7,11, 13, 17, 19, 23, 29.
ie 25- 7= 18
ans- C
ie 25- 7= 18
ans- C
Posts: 1
I get the logic, but when double checking on calculator I can divide 18! by 19 (n+1) even though 19 is prime.
18! = 6,402,373,705,728,000
18!/19 = 336,967,037,143,579
Posted from my mobile device
18! = 6,402,373,705,728,000
18!/19 = 336,967,037,143,579
Posted from my mobile device
fincrispe750
The calculator would have approximated it (and not given you decimals) because of the big numbers involved.
18! is not divisible by 19.
22! is not divisible by 23.
88! is not divisible by 89.
I can say this for sure because of the concept discussed here: https://anaprep.com/number-properties-r ... e-factors/
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