Azizakaria wrote:67. If k is an integer and (0.0025)(0.025)(0.00025) × 10^k is an integer, what is the
least possible value of k?
(A) −12
(B) −6
(C) 0
(D) 6
(E) 12

Another approach is to convert everything to fractions.

(0.0025)(0.025)(0.00025) × 10^k is an integer
So, (25/10,000)(25/1,000)(25/100,000) × 10^k is an integer
Simplify to get: (25³/1,000,000,000,000) × 10^k is an integer
To create an integer, we need 1,000,000,000,000 to EQUAL 10^k
So, k = 12

Answer: E

Cheers,
Brent

Azizakaria wrote:67. If k is an integer and (0.0025)(0.025)(0.00025) × 10^k is an integer, what is the
least possible value of k?
(A) −12
(B) −6
(C) 0
(D) 6
(E) 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 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.

Answer:E

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: E

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