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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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Question Stats:
63% (01:30) correct
37%
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based on 224
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If \(k\) is an integer and \((0.12)(0.013)(0.0014)(0.00015)10^k\) is an integer, what is the least possible value of \(k\)?
A. -14
B. -13
C. 0
D. 13
E. 14
A. -14
B. -13
C. 0
D. 13
E. 14
07 Apr 2024, 14:30
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Official Solution:
If \(k\) is an integer and \((0.12)(0.013)(0.0014)(0.00015)10^k\) is an integer, what is the least possible value of \(k\)?
A. -14
B. -13
C. 0
D. 13
E. 14
Don't make a mistake and assume that \(k\) must be 14, it's a trap answer! Notice that 12 and 15 when multiplied give \(180=18*10\) making the above equal to:
Therefore, the least possible value of \(k\), such that the above is an integer is 13.
Answer: D
If \(k\) is an integer and \((0.12)(0.013)(0.0014)(0.00015)10^k\) is an integer, what is the least possible value of \(k\)?
A. -14
B. -13
C. 0
D. 13
E. 14
\((0.12)(0.013)(0.0014)(0.00015)10^k=\)
\(=(12*10^{-2})(13*10^{-3})(14*10^{-4})(15*10^{-5})10^k=\)
\(=(12)(13)(14)(15)*10^{-14}*10^k\)
\(=(12*10^{-2})(13*10^{-3})(14*10^{-4})(15*10^{-5})10^k=\)
\(=(12)(13)(14)(15)*10^{-14}*10^k\)
Don't make a mistake and assume that \(k\) must be 14, it's a trap answer! Notice that 12 and 15 when multiplied give \(180=18*10\) making the above equal to:
\((18)(13)(14)*10*10^{-14}*10^k=\)
\(=(18)(13)(14)*10^{-13}*10^k\)
\(=(18)(13)(14)*10^{-13}*10^k\)
Therefore, the least possible value of \(k\), such that the above is an integer is 13.
Answer: D
General Discussion
29 Mar 2024, 21:08
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Bunuel
\((0.12)(0.013)(0.0014)(0.00015)10^k\)
\(\frac{12}{100} * \frac{13}{1000} * \frac{14}{10000} * \frac{15}{100000}\)
The value of \(k\) must be positive so that the 'zeros' in the denominator cancel out with the 'zeros' in the numerator. Hence, we can eliminate A, B, or C.
Observation 1: The multiplication of the values in the denominator results in \(10^{14}\)
Observation 2: The multiplication of 15 * 12 or 15 * 14 results in a value whose unit digit is zero. Hence, we need thirteen more zeros in the numerator.
Therefore the minimum value of \(k = 13\)
Option D
