Vamshi8411 wrote:
Which of the following is a terminating decimal, when expressed in decimals?
A. 17/223
B. 13/231
C. 41/3
D. 41/256
E. 35/324
Can we do this just by taking the unit digit in numerator and denominator?
Reduced fraction \(\frac{a}{b}\) (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal
if and only \(b\) (denominator) is of the form \(2^n5^m\), where \(m\) and \(n\) are non-negative integers. For example: \(\frac{7}{250}\) is a terminating decimal \(0.028\), as \(250\) (denominator) equals to \(2*5^2\). Fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and denominator \(10=2*5\).
BACK TO THE QUESTION:
\(\frac{41}{256}=\frac{41}{2^8}\), denominator has only prime factor 2 in its prime factorization, hence this fraction will be terminating decimal.
All other fractions (after reducing, if possible) have primes other than 2 and 5 in its prime factorization, hence they will be repeated decimals.
Answer: D.
For more check Number Theory chapter of Math Book:
math-number-theory-88376.htmlHope it helps..
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