Which of the following can divide 99999999 exactly?

Options

A. 9
B. 9999
C. 99
D. Each of (A), (B), and (C)

Answer

To determine which of the following numbers can divide 99999999 exactly, we need to analyze the factors of 99999999. Here’s a step-by-step solution:

### Step 1: Factor the number 99999999
To find the factors of 99999999, we can start by expressing it in a more manageable form. Notice that:

99999999 = 10^8 - 1

This can be factored using the difference of squares:

99999999 = (10^4 - 1)(10^4 + 1)

### Step 2: Factor 10^4 - 1
Now we can factor 10^4 - 1 further:

10^4 - 1 = (10^2 - 1)(10^2 + 1)

And we can factor 10^2 - 1:

10^2 - 1 = (10 - 1)(10 + 1) = 9 * 11

So, we have:

10^4 - 1 = (9)(11)(10^2 + 1)

Calculating 10^2 + 1 gives us:

10^2 + 1 = 100 + 1 = 101

Thus, we can write:

10^4 - 1 = 9 * 11 * 101

### Step 3: Factor 10^4 + 1
Next, we need to consider 10^4 + 1. This does not factor nicely over the integers, but we can keep it as is for now.

### Step 4: Combine the factors
Now we can combine our findings:

99999999 = (10^4 - 1)(10^4 + 1) = (9 * 11 * 101)(10^4 + 1)

### Step 5: Identify the divisors
From our factorization, we can see that the divisors of 99999999 include:

- 1
- 9
- 11
- 101
- 99 (which is 9 * 11)
- 909 (which is 9 * 101)
- 1111 (which is 11 * 101)
- 99999999 itself

### Conclusion
To determine which numbers can divide 99999999 exactly, we need to check the options provided in the question against the factors we have identified.