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14 Apr 2020, 06:23
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
41% (02:42) correct
59%
(02:34)
wrong
based on 119
sessions
History
Date
Time
Result
Not Attempted Yet
If N is a positive integer less than 31, how many values can N take so that (n + 1) is a factor of n! ?
A. 20
B. 18
C. 17
D. 16
E. 12
Are You Up For the Challenge: 700 Level Questions
A. 20
B. 18
C. 17
D. 16
E. 12
Are You Up For the Challenge: 700 Level Questions
14 Apr 2020, 06:44
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If n+1 is a prime number, n+1 is not a factor of n!. [n! will leave n as remainder when divided by n+1, if n+1 is a prime number]
Number of primes less than 31= 12
Number of values can n take so that (n + 1) is a factor of n! = 30-12=18
Number of primes less than 31= 12
Number of values can n take so that (n + 1) is a factor of n! = 30-12=18
Bunuel
14 Apr 2020, 06:47
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Bunuel
N < 31
We need to ensure that (n + 1) is a factor of n!
i.e. n+1 must be a composite number i.e .n+1 = prime can be simply avoided
Primes from 1 to 30 are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29,} = 10 Numbers
If n = 1, then (n + 1) is 2 which is NOT a factor of n! i.e. 1!
If n = 2, then (n + 1) is 3 which is NOT a factor of n! i.e. 2!
If n = 3, then (n + 1) is 4 which is NOT a factor of n! i.e. 3!
If n = 4, then (n + 1) is 5 which is NOT a factor of n! i.e. 4!
If n = 5, then (n + 1) is 6 which is a factor of n! i.e. 5!
If n = 6, then (n + 1) is 7 which is NOT a factor of n! i.e. 6!
If n = 7, then (n + 1) is 8 which is a factor of n! i.e. 7!
If n = 8, then (n + 1) is 9 which is a factor of n! i.e. 8!
If n = 9, then (n + 1) is 10 which is a factor of n! i.e. 9!
If n = 10, then (n + 1) is 11 which is NOT a factor of n! i.e. 10!
If n = 11, then (n + 1) is 12 which is NOT a factor of n! i.e. 11!
If n = 12, then (n + 1) is 13 which is NOT a factor of n! i.e. 12!
If n = 13, then (n + 1) is 14 which is a factor of n! i.e. 13!
If n = 14, then (n + 1) is 15 which is a factor of n! i.e. 14!
If n = 15, then (n + 1) is 16 which is a factor of n! i.e. 15!
If n = 16, then (n + 1) is 17 which is NOT a factor of n! i.e. 16!
If n = 17, then (n + 1) is 18 which is a factor of n! i.e. 17!
If n = 18, then (n + 1) is 19 which is NOT a factor of n! i.e. 18!
If n = 22, then (n + 1) is 23 which is NOT a factor of n! i.e. 22!
If n = 28, then (n + 1) is 29 which is NOT a factor of n! i.e. 28!
If n = 30, then (n + 1) is 31 which is NOT a factor of n! i.e. 30!
12 cases are the ones in which (n + 1) is NOT a factor of n!
i.e. 30-12 = 18 cases exist for whom (n + 1) is a factor of n!
Answer: Option B
