Problem
ALG-B1-M02-P027 A Product Became 78 Times Larger
In a product of five positive integers, each factor was decreased by \(4\). Could the new product become exactly \(78\) times the original one? If yes, find a suitable set of numbers.
Make four factors equal to \(1\), and denote the fifth by \(a\).
The point is not to guess all five numbers, but to create a large coefficient in front of one unknown. Let the numbers be \(1,1,1,1,a\). The original product is \(a\).
After decreasing each factor by \(4\), the product becomes \((-3)^4(a-4)=81a-324\).
We require
\[81a-324=78a.\]
Hence \(3a=324\), so \(a=108\). Check: the original product is \(108\), the new product is \((-3)^4\cdot104=8424\), and \(78\cdot108=8424\). Therefore \(1,1,1,1,108\) works.
The same architecture with different numbers: several small factors create a coefficient, then one linear condition remains.