A function, \(g\), has \(g(2) = 5\) and \(g(3) = 7\). In addition, \(g\) has the property that \[g(a) + g(b) = g(ab)\] for all positive
integers \(a\) and \(b\).
For example, \(g(6) = g(2) + g(3) =
12\).
What is the value of \(g(648)\)?
We can rewrite \(g(648)\) as: \[\begin{aligned} g(648) &=g(2\cdot 324)\\ &= g(2) + g(324)\\ &= g(2) + g(2\cdot 162)\\ &= g(2) + g(2) + g(162)\\ &= g(2) + g(2) + g(2\cdot 81)\\ &= g(2) + g(2) + g(2) + g(81)\\ &= g(2) + g(2) + g(2) + g(3\cdot 27)\\ &= g(2) + g(2) + g(2) + g(3) + g(27)\\ &= g(2) + g(2) + g(2) + g(3) + g(3\cdot 9)\\ &= g(2) + g(2) + g(2) + g(3) + g(3)+ g(9)\\ &= g(2) + g(2) + g(2) + g(3) + g(3)+ g(3\cdot 3)\\ &= g(2) + g(2) + g(2) + g(3) + g(3)+ g(3)+ g(3) \end{aligned}\]
Therefore, \(g(648) = 3g(2) + 4g(3)= 3(5) + 4(7) = 43.\)
Note:
While this answers the question, is there actually a function that
satisfies the requirements? The answer is yes.
One function that satisfies the requirements of the problem is the
function \(g\) defined by \[g(1) =0 \text{ and } g(2^p3^qr) = 5p +
7q\] for all non-negative integers \(p\) and \(q\) and all positive integers \(r\) that are not divisible by \(2\) or by \(3\). Can you see why this function
satisfies the requirements?