Question

Determine the largest value of \( n \) for which \[ 2^{n} \mid (75)! \]

• A. 71
• B. 72
• C. 73
• D. 74

Hint:

To find the highest power of a prime \(p\) dividing \(n!\), always use Legendre’s formula: \[ \sum \left\lfloor \frac{n}{p^k} \right\rfloor. \]

Answer 1

The Correct Option is B

Step 1: Understand the concept involved.
The highest power of \(2\) dividing \(75!\) is given by Legendre’s formula: \[ n = \left\lfloor \frac{75}{2} \right\rfloor + \left\lfloor \frac{75}{4} \right\rfloor + \left\lfloor \frac{75}{8} \right\rfloor + \left\lfloor \frac{75}{16} \right\rfloor + \left\lfloor \frac{75}{32} \right\rfloor + \left\lfloor \frac{75}{64} \right\rfloor. \]
Step 2: Evaluate each term.
\[ \left\lfloor \frac{75}{2} \right\rfloor = 37, \quad \left\lfloor \frac{75}{4} \right\rfloor = 18, \quad \left\lfloor \frac{75}{8} \right\rfloor = 9, \] \[ \left\lfloor \frac{75}{16} \right\rfloor = 4, \quad \left\lfloor \frac{75}{32} \right\rfloor = 2, \quad \left\lfloor \frac{75}{64} \right\rfloor = 1. \]
Step 3: Add all contributions.
\[ n = 37 + 18 + 9 + 4 + 2 + 1 = 71. \]
Step 4: Interpret the result.
This means the highest power of \(2\) dividing \(75!\) is \(2^{71}\).
Step 5: Final conclusion.
Hence, the largest value of \(n\) such that: \[ 2^{n} \mid (75)! \] is: \[ \boxed{71}. \]