If k is an integer and (.0025)(.025)(.00025)10^k is an integer, what is the least possible value of K?

(A) -12
(B) -6
(C) 0
(D) 6
(E) 12
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It should be 10^k, not 10k

If so, Ans: E

(0.0025)(0.025)(0.00025) × 10^k = 10^k/(2^6*10^6).

=> k=6+6=12

(.0025)(.025)(.00025)10^k = Integer

{25*10^(-4)}{25*10^(-3)}{25*10^(-5)}×10^k = Integer

(25^3)*10^(k-12) = Integer

I.e. (k-12) = Integer
I.e. Min (k-12)=0
I.e. (k)min = 12

Ans: Option E
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Hi. Can you please explain why it's not 10^-12? I understand how you get to 12, but have issues with how to know if it should be negative or positive.

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 to the decimal places of 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 result of 0.0025 x 0.025 x 0.00025 will have 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, making it an integer.

Answer: E
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