If `102!``=2^(alpha)*3^(beta)*5^(gamma)*7^(delta)`…, then

Options

A. (a) `alpha=98`
B. (b) `beta=2gamma+1`
C. (c) `alpha=2beta`
D. (d) `2gamma=3delta`

Explanation

To solve the problem of finding the values of α, β, γ, and δ in the expression \( 102! = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma} \cdot 7^{\delta} \cdots \), we will calculate the exponents of the prime factors in \( 102! \) using the formula for finding the exponent of a prime \( p \) in \( n! \):

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

### Step 1: Calculate α (Exponent of 2)

1. **Calculate \( \left\lfloor \frac{102}{2} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{2} \right\rfloor = 51 \]

2. **Calculate \( \left\lfloor \frac{102}{4} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{4} \right\rfloor = 25 \]

3. **Calculate \( \left\lfloor \frac{102}{8} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{8} \right\rfloor = 12 \]

4. **Calculate \( \left\lfloor \frac{102}{16} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{16} \right\rfloor = 6 \]

5. **Calculate \( \left\lfloor \frac{102}{32} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{32} \right\rfloor = 3 \]

6. **Calculate \( \left\lfloor \frac{102}{64} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{64} \right\rfloor = 1 \]

7. **Sum these values**:
\[ \alpha = 51 + 25 + 12 + 6 + 3 + 1 = 98 \]

### Step 2: Calculate β (Exponent of 3)

1. **Calculate \( \left\lfloor \frac{102}{3} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{3} \right\rfloor = 34 \]

2. **Calculate \( \left\lfloor \frac{102}{9} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{9} \right\rfloor = 11 \]

3. **Calculate \( \left\lfloor \frac{102}{27} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{27} \right\rfloor = 3 \]

4. **Calculate \( \left\lfloor \frac{102}{81} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{81} \right\rfloor = 1 \]

5. **Sum these values**:
\[ \beta = 34 + 11 + 3 + 1 = 49 \]

### Step 3: Calculate γ (Exponent of 5)

1. **Calculate \( \left\lfloor \frac{102}{5} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{5} \right\rfloor = 20 \]

2. **Calculate \( \left\lfloor \frac{102}{25} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{25} \right\rfloor = 4 \]

3. **Calculate \( \left\lfloor \frac{102}{125} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{125} \right\rfloor = 0 \]

4. **Sum these values**:
\[ \gamma = 20 + 4 + 0 = 24 \]

### Step 4: Calculate δ (Exponent of 7)

1. **Calculate \( \left\lfloor \frac{102}{7} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{7} \right\rfloor = 14 \]

2. **Calculate \( \left\lfloor \frac{102}{49} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{49} \right\rfloor = 2 \]

3. **Calculate \( \left\lfloor \frac{102}{343} \right\rfloor \)**:
\[ \left\lfloor \frac{102}{343} \right\rfloor = 0 \]

4. **Sum these values**:
\[ \delta = 14 + 2 + 0 = 16 \]

### Final Values
- \( \alpha = 98 \)
- \( \beta = 49 \)
- \( \gamma = 24 \)
- \( \delta = 16 \)

### Relationships
1. **Relation between β and γ**:
\[ \beta = 2 \cdot \gamma + 1 \quad \text{(since } 49 = 2 \cdot 24 + 1\text{)} \]

2. **Relation between α and β**:
\[ \alpha = 2 \cdot \beta \quad \text{(since } 98 = 2 \cdot 49\text{)} \]

3. **Relation between γ and δ**:
\[ 2 \cdot \gamma = 3 \cdot \delta \quad \text{(since } 48 = 3 \cdot 16\text{)} \]

### Summary
Thus, the values are:
- \( \alpha = 98 \)
- \( \beta = 49 \)
- \( \gamma = 24 \)
- \( \delta = 16 \)