If k is an integer and (0.0025)…
We are given the expression:
0.0025 x 0.025 x 0.00025 x 10^k = integer
To determine the least possible value of k, we want to use our rules of multiplication with decimals. When multiplying decimals, the final product has an equal number of decimal places that were originally in the numbers being multiplied. Let’s start by counting the number of decimal places.
0.0025 has 4 decimal places
0.025 has 3 decimal places
0.00025 has 5 decimal places
Thus, the product of 0.0025 x 0.025 x 0.00025 has 12 decimal places.
In order for 0.0025 x 0.025 x 0.00025 x 10^k = integer, k would have to be at least 12, since 10^12 times any number with 12 decimal places would move the decimal point of that number 12 places to the right.