67. If k is an integer and (0.0025)(0.025)(0.00025) × 10k is an integer, what is the
least possible value of k?
(A) −12
(B) −6
(C) 0
(D) 6
(E) 12
I don't understand why the right answer is E
I solved for A as (25 × 10^−4)(25 × 10^−3)(25 × 10^−5) × 10^k
thanks in advance
OG 16 Question 67
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- Azizakaria
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The OE for this problem has a typo.
The prompt should read as follows:
= (25)(10ˉ�)(25)(10ˉ³)(25)(10ˉ�)(10^k)
= 25³(10ˉ¹²)(10^k).
We can PLUG IN THE ANSWERS, which represent the least possible value of k.
A: 25³(10ˉ¹²)(10ˉ¹²) = 25³(10ˉ²�).
B: 25³(10ˉ¹²)(10ˉ�) = 25³(10ˉ¹�).
C: 25³(10ˉ¹²)(10�) = 25³(10ˉ¹²).
D: 25³(10ˉ¹²)(10�) = 25³(10ˉ�).
E: 25³(10ˉ¹²)(10¹²) = 25³(10�) = 25³.
Only E yields an integer value
The correct answer is E.
The prompt should read as follows:
(0.0025)(0.025)(0.00025) × 10^kAzizakaria 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
= (25)(10ˉ�)(25)(10ˉ³)(25)(10ˉ�)(10^k)
= 25³(10ˉ¹²)(10^k).
We can PLUG IN THE ANSWERS, which represent the least possible value of k.
A: 25³(10ˉ¹²)(10ˉ¹²) = 25³(10ˉ²�).
B: 25³(10ˉ¹²)(10ˉ�) = 25³(10ˉ¹�).
C: 25³(10ˉ¹²)(10�) = 25³(10ˉ¹²).
D: 25³(10ˉ¹²)(10�) = 25³(10ˉ�).
E: 25³(10ˉ¹²)(10¹²) = 25³(10�) = 25³.
Only E yields an integer value
The correct answer is E.
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Another approach is to convert everything to fractions.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
(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
- Max@Math Revolution
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Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.
If k is an integer and (0.0025)(0.025)(0.00025) X 10^k is an integer, what is the
least possible value of k?
(A) -12
(B) -6
(C) 0
(D) 6
(E) 12
(0.0025)(0.025)(0.00025) X 10^k =(25/10^4)*(25/10^3)*(25/10^5)*10^k.
(25/10^4)*(25/10^3)*(25/10^5) = (25^3)/(10^12).
That (25^3)/(10^12)*10^k is an integer implies that k should be more than or equal to 12. So the smallest number is 12.
If k is an integer and (0.0025)(0.025)(0.00025) X 10^k is an integer, what is the
least possible value of k?
(A) -12
(B) -6
(C) 0
(D) 6
(E) 12
(0.0025)(0.025)(0.00025) X 10^k =(25/10^4)*(25/10^3)*(25/10^5)*10^k.
(25/10^4)*(25/10^3)*(25/10^5) = (25^3)/(10^12).
That (25^3)/(10^12)*10^k is an integer implies that k should be more than or equal to 12. So the smallest number is 12.
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My approach:
We have 25/10000 * 25/1000 * 25/100000. To get rid of the denominators, we need 1000 * 10,000 * 100,000 in the numerator, or 10³ * 10� * 10�. Adding those exponents gives us (3 + 4 + 5), or 12, so we need 10 to the 12th power.
We have 25/10000 * 25/1000 * 25/100000. To get rid of the denominators, we need 1000 * 10,000 * 100,000 in the numerator, or 10³ * 10� * 10�. Adding those exponents gives us (3 + 4 + 5), or 12, so we need 10 to the 12th power.
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jmacym wrote:But there are they getting 25 ^ 3?????
From 25^3=5^6, if there is no 2, it can't turn into 10, which is meaningless.
That is, no matter how many 5 there are, it can't be 10 if there is no 2.
Hence, it is omitted.
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- fiza gupta
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(0.0025)(0.025)(0.00025) X 10^k
= [(25)(10ˉ�)(25)(10ˉ³)(25)(10ˉ�)](10^k)
= 25³(10ˉ¹²)(10^k)
K >=12
SO E
= [(25)(10ˉ�)(25)(10ˉ³)(25)(10ˉ�)](10^k)
= 25³(10ˉ¹²)(10^k)
K >=12
SO E
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We are given the expression: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
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
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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
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