If k is an integer and (0.0025)(0.025)(0.00025) × 10^k is an integer,

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

since 25 to the power any integer will always yield last digit as 5 and not yield any 0, k is just summation of number of places we need to move the decimal point to get each individual number to be an integer i.e. 4+3+5=12

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.

10^(-12) = 1/10^12

It's not 10^(-12) because if you substitute this value there you won't get an integer.

You can check easier example: 0.12*10^(-2) = 0.12*1/10^2=0.0012 but 0.12*10^2 = 12.

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

Another approach is to convert everything to fractions.

\((0.0025)(0.025)(0.00025)(10^k)\) is an integer

So, \((\frac{25}{10,000})(\frac{25}{1,000})(\frac{25}{100,000})(10^k)\) is an integer

Simplify to get: \((\frac{25^3}{1,000,000,000,000})(10^k)\) is an integer

To create an integer, we need \(1,000,000,000,000=10^k\)

So, \(k = 12\)

Answer: E

Cheers,
Brent

Simply, we will just count how many digits after decimal.

Here,

.0025=4
.025=3
.00025=5

Thus, 4+3+5=12 digits

And K must be 12 to get the least integer

Answer is E

Posted from my mobile device

Hi All,

We’re told that K is an integer and that (.0025)(.025)(.00025)(10^K) is an INTEGER. We’re asked for the least possible value of K. Since we’re given a lot of numbers to work with, this is essentially just an Arithmetic question – but you might find the work much faster to deal with depending on how you write-out the given information.

Based on how ‘spread out’ the Answers are written, there’s actually a great short-cut build into this prompt. Since we’re multiplying three fractional values together, that product will be a much smaller positive fraction. There’s no way to make (10^K) equal 0, so that piece of the product has to ‘offset’ all of the decimal places that would occur from multiplying those 3 fractional values together. Even without physically counting them up, we can see that there are a LOT of decimal places there – so there’s only one answer that could reasonably turn the overall product into an integer. If you count up the decimals, you’ll notice that there are 12 decimal places, which will also point you to the correct answer.

Barring those shortcuts, you could visualize the math by writing those decimals as fractions:

25/100 = .25
25/1000 = .025
25/10000 = .0025
Etc.

So we would have:

(25/10,000)(25/1,000)(25/100,000)(10^K)

We need to offset all 3 of those denominators, so that would require that we multiply by 10^4, 10^3 and 10^5, respectively… for a total of 10^12.

Final Answer:

GMAT Assassins aren’t born, they’re made,
Rich

Answer: Option E

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ScottTargetTestPrep

Thank you! I was just wondering, if you had 0.0025 X 0.0002, in addition to moving the decimal places over, wouldn't you have to factor in how the integers would interact together because you have an odd and an even number?

woohoo921 Interaction among the integers is significant if their product yields at least one factor of 10 (so it's not about odds or evens, it's about factors of 2 and 5).
I find it easier to convert all the decimals into fractions (with powers of 10 in the denominators). Then we can see whether the numerators are able to reduce any of the factors of 10 from the denominator, and incorporate that reduction into our answer.

The reason we just looked at the number of decimal places in this question is that we know the product does not have trailing zeros. In other words, if we were to express these decimals as fractions, the numerator would be 25 x 25 x 25 x 10^k and the denominator would be some power of 10. The only way that fraction could reduce to an integer is if all the 10s in the denominator cancel out, and since 25 x 25 x 25 does not contain any factors of 10, the 10s in the denominator must cancel with 10^k.

If the product were 0.0025 x 0.0002, then the product of 25 and 2 would cancel one 10 in the denominator, so the minimum value of k is one less than the number of decimal places in the two decimals.

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