`7^(9) +9^(7)` is divisible by

Options

A. 16
B. 24
C. 64
D. 72

Answer

To determine the divisibility of \( 7^9 + 9^7 \), we can use the Binomial Theorem. Let's break down the solution step by step.

### Step 1: Rewrite the expression using Binomial Expansion
We can express \( 7^9 \) and \( 9^7 \) in terms of a common base. We can write:
\[ 7^9 = (8 - 1)^9 \quad \text{and} \quad 9^7 = (8 + 1)^7 \]

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem, we can expand both expressions:
\[ (8 - 1)^9 = \sum_{k=0}^{9} \binom{9}{k} 8^{9-k} (-1)^k \]
\[ (8 + 1)^7 = \sum_{j=0}^{7} \binom{7}{j} 8^{7-j} (1)^j \]

### Step 3: Combine the expansions
Now, we combine the two expansions:
\[ 7^9 + 9^7 = \sum_{k=0}^{9} \binom{9}{k} 8^{9-k} (-1)^k + \sum_{j=0}^{7} \binom{7}{j} 8^{7-j} \]

### Step 4: Identify common terms
Notice that both expansions contain powers of \( 8 \). We will focus on the coefficients of \( 8^n \) from both expansions.

### Step 5: Calculate specific terms
- The constant term from \( (8 - 1)^9 \) is \( (-1)^9 = -1 \).
- The constant term from \( (8 + 1)^7 \) is \( 1 \).

### Step 6: Collect terms
The constant terms cancel out:
\[ (-1) + 1 = 0 \]
This indicates that the sum of the constant terms is zero.

### Step 7: Higher powers of 8
Now we need to look at the coefficients of \( 8^1 \) and \( 8^2 \):
- The coefficient of \( 8^1 \) in \( (8 - 1)^9 \) is \( \binom{9}{8}(-1) = -9 \).
- The coefficient of \( 8^1 \) in \( (8 + 1)^7 \) is \( \binom{7}{6} = 7 \).

Thus, the coefficient of \( 8^1 \) is:
\[ -9 + 7 = -2 \]

### Step 8: Coefficient of \( 8^2 \)
- The coefficient of \( 8^2 \) in \( (8 - 1)^9 \) is \( \binom{9}{7}(-1)^2 = 36 \).
- The coefficient of \( 8^2 \) in \( (8 + 1)^7 \) is \( \binom{7}{5} = 21 \).

Thus, the coefficient of \( 8^2 \) is:
\[ 36 + 21 = 57 \]

### Step 9: Final expression
Putting this all together, we have:
\[ 7^9 + 9^7 = 8^2 \cdot (k_1 + k_2) + \text{higher order terms} \]
where \( k_1 + k_2 \) is a combination of the coefficients we calculated.

### Conclusion
Since \( 7^9 + 9^7 \) contains \( 8^2 \) as a factor, it is divisible by \( 64 \).

Thus, the final answer is:
\[ \text{The expression } 7^9 + 9^7 \text{ is divisible by } 64. \]