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31 Dec 2023, 10:06
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Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If \(s\) is a positive integer and the ratio \(\frac{1}{s}\) is expressed as a decimal, is \(\frac{1}{s}\) a terminating decimal?
(1) \(s!\) ends with exactly one is 0.
(2) The sum of any two positive factors of \(s\) is even.
(1) \(s!\) ends with exactly one is 0.
(2) The sum of any two positive factors of \(s\) is even.
31 Dec 2023, 10:06
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Official Solution:
Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If \(s\) is a positive integer and the ratio \(\frac{1}{s}\) is expressed as a decimal, is \(\frac{1}{s}\) a terminating decimal?
A reduced fraction \(\frac{a}{b}\) (meaning that the fraction is already in its simplest form, so reduced to its lowest term) can be expressed as a terminating decimal if and only if the denominator \(b\) 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 the denominator \(250\) equals \(2*5^3\). The fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and the denominator \(10=2*5\).
Note that if the denominator already consists of only 2s and/or 5s, 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 a terminating decimal.
(We need to reduce the fraction in case the denominator has a prime other than 2 or 5, to see whether it can be reduced. For example, the fraction \(\frac{6}{15}\) has 3 as a prime in the denominator, and we need to know if it can be reduced.)
(1) \(s!\) ends with exactly one is 0.
This implies that \(s!\) has only one factor of 5, hence \(s\) can be 5, 6, 7, 8, or 9. If \(s\) is 5 or 8, \(\frac{1}{s}\) will be a terminating decimal. However, if \(s\) is 6, 7, or 9, \(\frac{1}{s}\) will NOT be a terminating decimal. Not sufficient.
(2) The sum of any two positive factors of \(s\) is even.
This implies that all factors of \(s\) are odd, so \(s\) itself is odd. If \(s\) is a power of 5: 5, 25, ... and so on, \(\frac{1}{s}\) will be a terminating decimal. However, if \(s\) is any other odd number (greater than 1), \(\frac{1}{s}\) will NOT be a terminating decimal. Not sufficient.
(1)+(2) Since from (2) \(s\) is odd, then from (1) it can be 5, 7, or 9. If \(s\) is 5, \(\frac{1}{s}\) will be a terminating decimal. However, if \(s\) is 7, or 9, \(\frac{1}{s}\) will NOT be a terminating decimal. Not sufficient.
Answer: E
Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If \(s\) is a positive integer and the ratio \(\frac{1}{s}\) is expressed as a decimal, is \(\frac{1}{s}\) a terminating decimal?
A reduced fraction \(\frac{a}{b}\) (meaning that the fraction is already in its simplest form, so reduced to its lowest term) can be expressed as a terminating decimal if and only if the denominator \(b\) 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 the denominator \(250\) equals \(2*5^3\). The fraction \(\frac{3}{30}\) is also a terminating decimal, as \(\frac{3}{30}=\frac{1}{10}\) and the denominator \(10=2*5\).
Note that if the denominator already consists of only 2s and/or 5s, 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 a terminating decimal.
(We need to reduce the fraction in case the denominator has a prime other than 2 or 5, to see whether it can be reduced. For example, the fraction \(\frac{6}{15}\) has 3 as a prime in the denominator, and we need to know if it can be reduced.)
(1) \(s!\) ends with exactly one is 0.
This implies that \(s!\) has only one factor of 5, hence \(s\) can be 5, 6, 7, 8, or 9. If \(s\) is 5 or 8, \(\frac{1}{s}\) will be a terminating decimal. However, if \(s\) is 6, 7, or 9, \(\frac{1}{s}\) will NOT be a terminating decimal. Not sufficient.
(2) The sum of any two positive factors of \(s\) is even.
This implies that all factors of \(s\) are odd, so \(s\) itself is odd. If \(s\) is a power of 5: 5, 25, ... and so on, \(\frac{1}{s}\) will be a terminating decimal. However, if \(s\) is any other odd number (greater than 1), \(\frac{1}{s}\) will NOT be a terminating decimal. Not sufficient.
(1)+(2) Since from (2) \(s\) is odd, then from (1) it can be 5, 7, or 9. If \(s\) is 5, \(\frac{1}{s}\) will be a terminating decimal. However, if \(s\) is 7, or 9, \(\frac{1}{s}\) will NOT be a terminating decimal. Not sufficient.
Answer: E
