Question:
What is the least positive integer that is not a factor of \( 25! \) and is not a prime number?
What is the least positive integer that is not a factor of \( 25! \) and is not a prime number?
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To find numbers not dividing a factorial, test integers just greater than the factorial’s base number. Factorials cover all smaller integers.
Updated On: Oct 7, 2025
- 26
- 28
- 36
- 56
- 58
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The Correct Option is B
Solution and Explanation
Step 1: Factors of \( 25! \).
Since \( 25! \) is the product of integers from 1 to 25, every integer up to 25 divides \( 25! \).
Step 2: Numbers greater than 25.
We seek the smallest integer greater than 25 that is not a factor of \( 25! \) and is not prime.
Step 3: Check candidates.
- 26 = \( 2 \times 13 \). Divides \( 25! \). Not correct.
- 27 = \( 3^3 \). Divides \( 25! \). Not correct.
- 28 = \( 2^2 \times 7 \). This does not divide \( 25! \) fully, since 28>25 and is not prime. Hence, 28 is the least.
Step 4: Conclusion.
Thus, the required integer is: \[ \boxed{\text{(B) 28}} \]
Since \( 25! \) is the product of integers from 1 to 25, every integer up to 25 divides \( 25! \).
Step 2: Numbers greater than 25.
We seek the smallest integer greater than 25 that is not a factor of \( 25! \) and is not prime.
Step 3: Check candidates.
- 26 = \( 2 \times 13 \). Divides \( 25! \). Not correct.
- 27 = \( 3^3 \). Divides \( 25! \). Not correct.
- 28 = \( 2^2 \times 7 \). This does not divide \( 25! \) fully, since 28>25 and is not prime. Hence, 28 is the least.
Step 4: Conclusion.
Thus, the required integer is: \[ \boxed{\text{(B) 28}} \]
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