What is the greatest integer among the following by which the number `5^(5) + 7^(5)` is divisible?
(a)6
(b)8
(c)11
(d)12

Options

A. 6
B. 8
C. 11
D. 12

Answer

To solve the problem of finding the greatest integer among the options by which the number \( 5^5 + 7^5 \) is divisible, we can follow these steps:

### Step 1: Identify the expression
We need to evaluate the expression \( 5^5 + 7^5 \).

### Step 2: Recognize the pattern
We notice that we have a sum of two terms raised to an odd power, specifically \( n = 5 \) (which is odd). There is a mathematical property that states:
- If \( n \) is odd, then \( a^n + b^n \) is divisible by \( a + b \).

### Step 3: Apply the property
In our case, we can set:
- \( a = 5 \)
- \( b = 7 \)

Thus, we can calculate:
\[ a + b = 5 + 7 = 12 \]

### Step 4: Conclusion
Since \( 5^5 + 7^5 \) is divisible by \( 12 \), we check the options provided:
- (a) 6
- (b) 8
- (c) 11
- (d) 12

Among these options, \( 12 \) is the greatest integer that divides \( 5^5 + 7^5 \).

### Final Answer
The greatest integer among the options by which \( 5^5 + 7^5 \) is divisible is **12** (option d).

---