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If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?
1) s is a factor of 100
2) r is a factor of 100
I don't understand the answer explanation in the
OG, could someone please explain in detail?
THEORY: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 \(2*5^3\). Fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and denominator \(10=2*5\).
Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.For example \(\frac{x}{2^n5^m}\), (where x, n and m are integers) will always be terminating decimal.
(We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction \(\frac{6}{15}\) has 3 as prime in denominator and we need to know if it can be reduced.)
Questions testing this concept:
does-the-decimal-equivalent-of-p-q-where-p-and-q-are-89566.htmlany-decimal-that-has-only-a-finite-number-of-nonzero-digits-101964.htmlif-a-b-c-d-and-e-are-integers-and-p-2-a3-b-and-q-2-c3-d5-e-is-p-q-a-terminating-decimal-125789.html700-question-94641.htmlis-r-s2-is-a-terminating-decimal-91360.htmlpl-explain-89566.htmlwhich-of-the-following-fractions-88937.htmlBACK TO THE ORIGINAL QUESTION:If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?(1) s is a factor of 100. Factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50 and 100. All these numbers are of the form \(2^n5^m\) (for example 1=2^0*5^0, 2=2^1*5^0, ...), therefore no matter what is the value of r, r/s will always will be terminating decimal. Sufficient.
(2) r is a factor of 100. We need to know about the denominator. Not sufficient.
Answer: A.
Hope it's clear.