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16 Sep 2014, 01:45
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Set A consists of all distinct prime numbers which are 2 more than a multiple of 3. If set B consists of distinct integers, is set B a subset of set A?
(1) Set B consists of two positive integers whose product is 10.
(2) The product of the reciprocals of all elements in set B is a terminating decimal.
(1) Set B consists of two positive integers whose product is 10.
(2) The product of the reciprocals of all elements in set B is a terminating decimal.
01 Jun 2016, 22:51
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glochou
THEORY:
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.)
Check Terminating and Recurring Decimals Problems in our Special Questions Directory.
16 Sep 2014, 01:45
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Official Solution:
Set A consists of all distinct prime numbers which are 2 more than a multiple of 3. If set B consists of distinct integers, is set B a subset of set A?
According to the stem, set A = {2, 5, 11, 17, 23, 29, 41, ...}
(1) Set B consists of two positive integers whose product is 10.
10 can be expressed as the product of two integers in two ways: 10 = 1*10 and 10 = 2*5. If set B = {1, 10}, then the answer to the question is NO, but if set B = {2, 5}, then the answer to the question is YES. Not sufficient.
(2) The product of the reciprocals of all elements in set B is a terminating decimal.
If set B = {4, 8}, then the answer to the question is NO, but if set B = {2, 5}, then the answer to the question is YES. Not sufficient.
(1)+(2) Both possible sets from (1) satisfy the condition stated in the second statement: \( \frac{1}{1}*\frac{1}{10}=\frac{1}{10}=0.1\) (a terminating decimal) and \(\frac{1}{2}*\frac{1}{5}=\frac{1}{10}=0.1\) (a terminating decimal). Thus, we still have two sets, which give two different answers to the question. Not sufficient.
Answer: E
Set A consists of all distinct prime numbers which are 2 more than a multiple of 3. If set B consists of distinct integers, is set B a subset of set A?
According to the stem, set A = {2, 5, 11, 17, 23, 29, 41, ...}
(1) Set B consists of two positive integers whose product is 10.
10 can be expressed as the product of two integers in two ways: 10 = 1*10 and 10 = 2*5. If set B = {1, 10}, then the answer to the question is NO, but if set B = {2, 5}, then the answer to the question is YES. Not sufficient.
(2) The product of the reciprocals of all elements in set B is a terminating decimal.
If set B = {4, 8}, then the answer to the question is NO, but if set B = {2, 5}, then the answer to the question is YES. Not sufficient.
(1)+(2) Both possible sets from (1) satisfy the condition stated in the second statement: \( \frac{1}{1}*\frac{1}{10}=\frac{1}{10}=0.1\) (a terminating decimal) and \(\frac{1}{2}*\frac{1}{5}=\frac{1}{10}=0.1\) (a terminating decimal). Thus, we still have two sets, which give two different answers to the question. Not sufficient.
Answer: E
