M27-08

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