If (150! - 1) is divisible by a prime number x, then which of the following is true?
A. x = 2
B. x = 3
C. 3 < x ≤ 150
D. x > 150
Answer (2)
The prime number x that divides 150 ! − 1 cannot be 2 or 3, as shown through direct evaluation. Thus, the correct option is D, indicating that x must be greater than 150. While it is possible for x to fall within 3 to 150, primes in that range do not meet the division criteria.
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To determine which statement is true regarding the divisibility of ( 150 ! − 1 ) by a prime number x , let's consider the nature of factorials and divisibility:
Definition of Factorial :
A factorial, represented by n ! , is the product of all positive integers up to n . For example, 5 ! = 5 × 4 × 3 × 2 × 1 = 120 .
Thus, 150 ! = 150 × 149 × … × 3 × 2 × 1 .
Divisibility by Prime Numbers :
Any number n ! is divisible by all integers from 1 to n because it is the product of these numbers.
Therefore, 150 ! is divisible by all prime numbers less than or equal to 150.
(150! - 1) Analysis :
150 ! , being divisible by all prime numbers up to 150, means that when we subtract 1 from 150 ! , the result 150 ! − 1 is exactly 1 less than a multiple of any of those primes.
Thus, 150 ! − 1 cannot be divisible by any prime number x that is less than or equal to 150 because it is 1 less than a multiple of such a prime number.
Conclusion :
The only possibility left is that any prime x which divides 150 ! − 1 must be greater than 150.
Therefore, the correct answer is (D) 150"> x > 150 . This is the only condition where x might divide 150 ! − 1 because it is not a divisor of 150 ! and hence potentially could divide 150 ! − 1 .
