if Kis the smallest positive integer such that(2k)(5200) has 303digits.

Find the sum of the digits of that number when expanded.

How do you "expand" a number?

Is (2k)5200 a single number or a misprint? Why write it that way?

To find the smallest positive integer 'K' such that (2k)(5200) has 303 digits, we need to find the value of 'K' that makes the product (2k)(5200) as close as possible to a number with 303 digits.

Let's break down the problem into steps:

Step 1: Find the number of digits in (2k)(5200):
The number of digits in a positive integer 'N' can be found using the logarithmic function. Specifically, the number of digits in 'N' is given by ⌊log₁₀(N)⌋ + 1.

In this case, the number of digits in (2k)(5200) is 303. So we have:
⌊log₁₀((2k)(5200))⌋ + 1 = 303

Step 2: Simplify the equation:
Simplify the equation by removing the floor function:
log₁₀((2k)(5200)) + 1 = 303

Subtract 1 from both sides:
log₁₀((2k)(5200)) = 302

Step 3: Solve for 'k':
Let's combine the logarithmic expression into exponents to isolate 'k':
10^(log₁₀((2k)(5200))) = 10^302

Simplify further:
(2k)(5200) = 10^302

Divide both sides by 5200 to isolate 'k':
2k = 10^302 / 5200

Divide both sides by 2 to solve for 'k':
k = (10^302) / (5200 * 2)

Step 4: Find the sum of the digits of the number when expanded:
To find the sum of the digits of a number, we need to add each digit individually. Let's calculate the value of (2k)(5200) and find its digit sum.

Multiply the value of 'k' obtained in Step 3 by 5200 and 2:
Number = (k)(5200)(2)

Now, calculate the value of 'Number' and find the sum of its digits.