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07 Jun 2015, 08:53
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T is the set of all positive integers \(x\) such that \(x^2\) is a multiple of both 27 and 375. Which of the following integers must be a factor of every integer \(x\) in T?
I. 9
II. 15
III. 27
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
I. 9
II. 15
III. 27
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
07 Jun 2015, 08:53
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Official Solution:
T is the set of all positive integers \(x\) such that \(x^2\) is a multiple of both 27 and 375. Which of the following integers must be a factor of every integer \(x\) in T?
I. 9
II. 15
III. 27
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
The least common multiple of \(27 = 3^3\) and \(375 = 3*5^3\) is \(3^3*5^3\).
For the integer \(x\), the minimum value of \(x^2\) that can be a multiple of \(3^3*5^3\) is \(3^4*5^4\). Hence, the smallest possible value for \(x\) is \(3^2*5^2\). Therefore:
\(T = \{3^2*5^2, 2*(3^2*5^2), 3*(3^2*5^2), 4*(3^2*5^2), ...\}\).
The question asks which of the options must be a factor of every integer \(x\) in T. Only options I and II meet this criterion because \(27 = 3^3\), option III, is not a factor of \(3^2*5^2\).
Answer: C
T is the set of all positive integers \(x\) such that \(x^2\) is a multiple of both 27 and 375. Which of the following integers must be a factor of every integer \(x\) in T?
I. 9
II. 15
III. 27
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
The least common multiple of \(27 = 3^3\) and \(375 = 3*5^3\) is \(3^3*5^3\).
For the integer \(x\), the minimum value of \(x^2\) that can be a multiple of \(3^3*5^3\) is \(3^4*5^4\). Hence, the smallest possible value for \(x\) is \(3^2*5^2\). Therefore:
\(T = \{3^2*5^2, 2*(3^2*5^2), 3*(3^2*5^2), 4*(3^2*5^2), ...\}\).
The question asks which of the options must be a factor of every integer \(x\) in T. Only options I and II meet this criterion because \(27 = 3^3\), option III, is not a factor of \(3^2*5^2\).
Answer: C
General Discussion
