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The function f(n) is defined as the product of all integers from 1 to n, inclusive, and the function g(n) is defined as the product of all odd integers from 1 to n, inclusive, where n is a positive integer. If p is a prime factor of {f(150)/g(150)} + 1, then which of the following must be true?
- a)p < 10
- b)10 < p < 25
- c)25 < p < 50
- d)50 < p < 75
- e)p > 75
Correct answer is option 'E'. Can you explain this answer?
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The function f(n) is defined as the product of all integers from 1 to ...
Understanding the Functions f(n) and g(n)
The function f(n) is defined as:
- f(n) = n! (the factorial of n)
The function g(n) is defined as:
- g(n) = product of all odd integers from 1 to n
For example, for n = 150:
- f(150) = 150!
- g(150) includes odd numbers: 1, 3, 5, ..., 149.
Calculating f(150) / g(150)
To find f(150) / g(150), we can express g(n):
- g(n) = (1 * 3 * 5 * ... * n) = (n! / (2^k * k!)) where k is the number of odd integers.
For n = 150:
- k = 75 (as there are 75 odd integers between 1 and 150).
- Thus, g(150) = (150! / (2^75 * 75!)).
Now, we can rewrite f(150) / g(150):
- f(150) / g(150) = 150! / [(150! / (2^75 * 75!))] = 2^75 * 75!.
Analyzing {f(150) / g(150)} + 1
Now we compute:
- {f(150) / g(150)} + 1 = 2^75 * 75! + 1.
This expression is crucial for determining the prime factors.
Finding Prime Factors
We need to analyze the expression \(2^{75} \cdot 75! + 1\):
- Since \(2^{75} \cdot 75!\) is even, \(2^{75} \cdot 75! + 1\) is odd.
- Any prime factor of \(2^{75} \cdot 75! + 1\) cannot be 2.
Next, we check the largest prime factor of \(75!\):
- The largest prime less than 75 is 73.
- However, since we add 1, we need to consider primes greater than 75.
Thus, the result concludes that:
Conclusion
- The prime factor \(p\) of \(f(150) / g(150) + 1\) must be greater than 75, confirming option E.
The function f(n) is defined as:
- f(n) = n! (the factorial of n)
The function g(n) is defined as:
- g(n) = product of all odd integers from 1 to n
For example, for n = 150:
- f(150) = 150!
- g(150) includes odd numbers: 1, 3, 5, ..., 149.
Calculating f(150) / g(150)
To find f(150) / g(150), we can express g(n):
- g(n) = (1 * 3 * 5 * ... * n) = (n! / (2^k * k!)) where k is the number of odd integers.
For n = 150:
- k = 75 (as there are 75 odd integers between 1 and 150).
- Thus, g(150) = (150! / (2^75 * 75!)).
Now, we can rewrite f(150) / g(150):
- f(150) / g(150) = 150! / [(150! / (2^75 * 75!))] = 2^75 * 75!.
Analyzing {f(150) / g(150)} + 1
Now we compute:
- {f(150) / g(150)} + 1 = 2^75 * 75! + 1.
This expression is crucial for determining the prime factors.
Finding Prime Factors
We need to analyze the expression \(2^{75} \cdot 75! + 1\):
- Since \(2^{75} \cdot 75!\) is even, \(2^{75} \cdot 75! + 1\) is odd.
- Any prime factor of \(2^{75} \cdot 75! + 1\) cannot be 2.
Next, we check the largest prime factor of \(75!\):
- The largest prime less than 75 is 73.
- However, since we add 1, we need to consider primes greater than 75.
Thus, the result concludes that:
Conclusion
- The prime factor \(p\) of \(f(150) / g(150) + 1\) must be greater than 75, confirming option E.
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The function f(n) is defined as the product of all integers from 1 to n, inclusive, and the function g(n) is defined as the product of all odd integers from 1 to n, inclusive, where n is a positive integer. If p is a prime factor of {f(150)/g(150)} + 1, then which of the following must be true?a)p < 10b)10 < p < 25c)25 < p < 50d)50 < p < 75e)p > 75Correct answer is option 'E'. Can you explain this answer?
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