Determine the largest 2-digit prime factor of the integer `((200),(100))` i.e., `""^(200)C_(100)`

Options

A. 61
B. 59
C. 17
D. 20

Answer

To determine the largest 2-digit prime factor of the binomial coefficient \( \binom{200}{100} \), we can follow these steps:

### Step 1: Understand the Binomial Coefficient
The binomial coefficient \( \binom{n}{r} \) can be expressed as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
For our case, we have:
\[ \binom{200}{100} = \frac{200!}{100! \times 100!} \]

### Step 2: Identify Prime Factors
To find the prime factors of \( \binom{200}{100} \), we need to consider the prime factorization of the numerator and denominator. A prime \( p \) will contribute to \( \binom{200}{100} \) if it appears more times in the numerator than in the denominator.

### Step 3: Count the Occurrences of Prime \( p \)
Using the formula for the exponent of a prime \( p \) in \( n! \):
\[ e_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor \]
We can calculate the exponent of \( p \) in \( \binom{200}{100} \):
\[ e_p\left(\binom{200}{100}\right) = e_p(200!) - 2 \cdot e_p(100!) \]

### Step 4: Find 2-Digit Prime Factors
The 2-digit prime numbers are: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

We will check each of these primes to see if they divide \( \binom{200}{100} \) and find the largest one.

### Step 5: Check Each Prime
1. **For \( p = 61 \)**:
\[ e_{61}(200!) = \left\lfloor \frac{200}{61} \right\rfloor = 3 \]
\[ e_{61}(100!) = \left\lfloor \frac{100}{61} \right\rfloor = 1 \]
\[ e_{61}\left(\binom{200}{100}\right) = 3 - 2 \cdot 1 = 1 \]
(Contributes)

2. **For \( p = 59 \)**:
\[ e_{59}(200!) = \left\lfloor \frac{200}{59} \right\rfloor = 3 \]
\[ e_{59}(100!) = \left\lfloor \frac{100}{59} \right\rfloor = 1 \]
\[ e_{59}\left(\binom{200}{100}\right) = 3 - 2 \cdot 1 = 1 \]
(Contributes)

3. **For \( p = 53 \)**:
\[ e_{53}(200!) = \left\lfloor \frac{200}{53} \right\rfloor = 3 \]
\[ e_{53}(100!) = \left\lfloor \frac{100}{53} \right\rfloor = 1 \]
\[ e_{53}\left(\binom{200}{100}\right) = 3 - 2 \cdot 1 = 1 \]
(Contributes)

4. **Continue checking down to 11**:
- All primes larger than 61 will not contribute as they will not appear in \( 100! \).

### Step 6: Conclusion
After checking all 2-digit primes, the largest 2-digit prime factor of \( \binom{200}{100} \) is **61**.