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16 Sep 2014, 01:27
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The product of three distinct positive integers equals the square of the largest of these three numbers. What is the product of the two smaller numbers?
(1) The average (arithmetic mean) of the three numbers is \(\frac{34}{3}\).
(2) The largest of the three numbers is 24.
(1) The average (arithmetic mean) of the three numbers is \(\frac{34}{3}\).
(2) The largest of the three numbers is 24.
16 Sep 2014, 01:27
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Official Solution:
The product of three distinct positive integers equals the square of the largest of these three numbers. What is the product of the two smaller numbers?
This is a 750 level question.
Let's denote the three integers as \(a\), \(b\), and \(c\), with \(0 < a < b < c\). Given: \(abc=c^2\), thus, \(ab=c\). The question is: what is the value of \(ab=c\)?
(1) The average (arithmetic mean) of the three numbers is \(\frac{34}{3}\).
This essentially states that \(a+b+c=34\). Replace \(c\) with \(ab\) to yield \(a+b+ab=34\). From this, we can infer that \((a+1)(b+1)=35\). Given \(a\) and \(b\) are integers, this implies \(a+1=5\) and \(b+1=7\), leading to \(a=4\) and \(b=6\). Therefore, \(ab=24\). This is sufficient. (Note that the scenario where \(a+1=1\) and \(b+1=35\) isn't valid, as it results in \(a=0\), contradicting the condition that all integers are positive).
(2) The largest of the three numbers is 24.
This statement directly provides the value of \(c\), which is sufficient.
Answer: D
The product of three distinct positive integers equals the square of the largest of these three numbers. What is the product of the two smaller numbers?
This is a 750 level question.
Let's denote the three integers as \(a\), \(b\), and \(c\), with \(0 < a < b < c\). Given: \(abc=c^2\), thus, \(ab=c\). The question is: what is the value of \(ab=c\)?
(1) The average (arithmetic mean) of the three numbers is \(\frac{34}{3}\).
This essentially states that \(a+b+c=34\). Replace \(c\) with \(ab\) to yield \(a+b+ab=34\). From this, we can infer that \((a+1)(b+1)=35\). Given \(a\) and \(b\) are integers, this implies \(a+1=5\) and \(b+1=7\), leading to \(a=4\) and \(b=6\). Therefore, \(ab=24\). This is sufficient. (Note that the scenario where \(a+1=1\) and \(b+1=35\) isn't valid, as it results in \(a=0\), contradicting the condition that all integers are positive).
(2) The largest of the three numbers is 24.
This statement directly provides the value of \(c\), which is sufficient.
Answer: D
General Discussion
10 Aug 2015, 18:37
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Bunuel,
How did you get from a+b+ab=34 to (a+1)(b+1)=35?
Tks,
How did you get from a+b+ab=34 to (a+1)(b+1)=35?
Tks,
