Question:
For how many values of ‘n < 50’ are possible, such that n! is divisible by n2.
For how many values of ‘n < 50’ are possible, such that n! is divisible by n2.
Updated On: Sep 10, 2024
- 31
- 32
- 33
- 34
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The Correct Option is B
Solution and Explanation
It is known that, n! is always divisible by ‘n’.
For n! to be divisible by n2, (n - 1)! must be divisible by n.
For any prime number (n - 1)! is not divisible by n and also for n = 1, 4.
So, the possible values are 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24,25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49 = 32 numbers
Or 49 - number of primes under 50 - 2 = 49 - 15 - 2 = 32 numbers
So, the correct option is (B) : 32.
For n! to be divisible by n2, (n - 1)! must be divisible by n.
For any prime number (n - 1)! is not divisible by n and also for n = 1, 4.
So, the possible values are 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24,25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49 = 32 numbers
Or 49 - number of primes under 50 - 2 = 49 - 15 - 2 = 32 numbers
So, the correct option is (B) : 32.
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