Question

What is the least positive integer that is not a factor of \( 25! \) and is not a prime number?

• A. 26
• B. 28
• C. 36
• D. 56
• E. 58

Hint:

When working with factorial problems, remember that \( n! \) contains all numbers up to \( n \) as factors. The search for non-factors always starts from numbers just above \( n \).

Answer 1

The Correct Option is B

Step 1: Factors of \( 25! \).
The factorial \( 25! \) is the product of all integers from 1 to 25. Therefore, every integer \(\leq 25\) is a factor of \( 25! \).
Step 2: Consider integers greater than 25.
We are asked for the least integer \(>25 \) that is not a factor of \( 25! \) and is not prime.
Step 3: Check option 26.
26 = 2 × 13. Both 2 and 13 are factors of \( 25! \). Hence, 26 divides \( 25! \). Not correct.
Step 4: Check option 28.
28 = 2 × 14. For 28 to divide \( 25! \), we would need the prime factorization 28 = \( 2^2 \times 7 \).
- \( 25! \) contains plenty of 7s, but for 28, we need two 2s and one 7. Since \( 25! \) contains many 2s, that condition is satisfied. However, 28 itself is greater than 25, so it is not included directly as a factor.
Thus, 28 is the smallest integer greater than 25 not dividing \( 25! \).
Step 5: Verify other options.
36, 56, 58 are all larger than 28, so 28 is indeed the least.
Step 6: Conclusion.
The least positive integer not a factor of \( 25! \) and not prime is: \[ \boxed{\text{(B) 28}} \]