Find the highest power of 2 in 50!

Answer

To find the highest power of 2 in 50!, we can use the formula for determining the highest power of a prime \( p \) that divides \( n! \):

\[ \text{Highest power of } p \text{ in } n! = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]

In this case, \( n = 50 \) and \( p = 2 \).

### Step-by-Step Solution:

1. **Calculate \( \left\lfloor \frac{50}{2^1} \right\rfloor \)**:
\[ \left\lfloor \frac{50}{2} \right\rfloor = \left\lfloor 25 \right\rfloor = 25 \]

2. **Calculate \( \left\lfloor \frac{50}{2^2} \right\rfloor \)**:
\[ \left\lfloor \frac{50}{4} \right\rfloor = \left\lfloor 12.5 \right\rfloor = 12 \]

3. **Calculate \( \left\lfloor \frac{50}{2^3} \right\rfloor \)**:
\[ \left\lfloor \frac{50}{8} \right\rfloor = \left\lfloor 6.25 \right\rfloor = 6 \]

4. **Calculate \( \left\lfloor \frac{50}{2^4} \right\rfloor \)**:
\[ \left\lfloor \frac{50}{16} \right\rfloor = \left\lfloor 3.125 \right\rfloor = 3 \]

5. **Calculate \( \left\lfloor \frac{50}{2^5} \right\rfloor \)**:
\[ \left\lfloor \frac{50}{32} \right\rfloor = \left\lfloor 1.5625 \right\rfloor = 1 \]

6. **Calculate \( \left\lfloor \frac{50}{2^6} \right\rfloor \)**:
\[ \left\lfloor \frac{50}{64} \right\rfloor = \left\lfloor 0.78125 \right\rfloor = 0 \]

Since \( \left\lfloor \frac{50}{2^6} \right\rfloor \) is 0, we stop here.

7. **Add all the results together**:
\[ 25 + 12 + 6 + 3 + 1 = 47 \]

Thus, the highest power of 2 in \( 50! \) is **47**.