If k is integer, and if `0.02468xx10^(k)` is greater than 10,000, what is the least possible value of k?

Answer

To solve the problem, we need to find the least integer value of \( k \) such that:

\[ 0.02468 \times 10^k > 10,000 \]

### Step 1: Rewrite the inequality
We start by rewriting the inequality:

\[ 0.02468 \times 10^k > 10,000 \]

### Step 2: Convert 10,000 to scientific notation
Next, we can express 10,000 in scientific notation:

\[ 10,000 = 10^4 \]

### Step 3: Set up the inequality with scientific notation
Now we can rewrite our inequality as:

\[ 0.02468 \times 10^k > 10^4 \]

### Step 4: Isolate \( 10^k \)
To isolate \( 10^k \), we divide both sides of the inequality by \( 0.02468 \):

\[ 10^k > \frac{10^4}{0.02468} \]

### Step 5: Calculate \( \frac{10^4}{0.02468} \)
Now we need to calculate \( \frac{10^4}{0.02468} \):

\[ \frac{10^4}{0.02468} = \frac{10000}{0.02468} \approx 405,000.405 \]

### Step 6: Take logarithm to find \( k \)
To find \( k \), we can take the logarithm (base 10) of both sides:

\[ k > \log_{10}(405,000.405) \]

Using a calculator, we find:

\[ \log_{10}(405,000.405) \approx 5.608 \]

### Step 7: Determine the least integer value of \( k \)
Since \( k \) must be an integer, we round up from \( 5.608 \) to the next whole number:

\[ k \geq 6 \]

Thus, the least possible value of \( k \) is:

\[ \boxed{6} \]