Maximizing (a^2b^4) in the Equation ( frac{2}{bc} frac{4}{ab} frac{3}{ac} frac{18}{abc} )

In this article, we will explore a problem involving the optimization of a mathematical equation with positive integers. The goal is to maximize the expression (a^2 b^4) given the equation:

( frac{2}{bc} frac{4}{ab} frac{3}{ac} frac{18}{abc} )

Solving the Equation

To solve the equation, we first find a common denominator on the left side, which is (abc). Rewriting each term with this common denominator, we get:

( frac{2a}{abc} frac{4c}{abc} frac{3b}{abc} frac{18}{abc} )

Combining the left-hand side gives:

( frac{2a 4c 3b}{abc} frac{18}{abc} )

Eliminating the common denominator (abc) since (a), (b), and (c) are positive integers and (abc eq 0), we obtain:

( 2a 4c 3b 18 )

Maximizing (a^2 b^4)

Our objective is to maximize the expression (a^2 b^4). To do this, we express (c) in terms of (a) and (b):

( 4c 18 - 2a - 3b )

Therefore,

( c frac{18 - 2a - 3b}{4} )

For (c) to be a positive integer, the expression

( 18 - 2a - 3b 0 )

and

( 18 - 2a - 3b equiv 0 pmod{4} )

Step 1: Finding Integer Solutions for (a) and (b)

From the inequality:

( 2a 3b 18 )

Step 2: Analyzing the Condition

To simplify the condition ( 18 - 2a - 3b equiv 0 pmod{4} ), we note:

( 18 equiv 2 pmod{4} )

Therefore, we need:

( 2 - 2a - 3b equiv 0 pmod{4} )

or equivalently,

( 2a 3b equiv 2 pmod{4} )

This means (2a 3b) must be of the form (2 4k), where (k) is an integer.

Step 3: Solving the Inequality

Let’s consider different values for (a) and check the corresponding values for (b):

(a) (2a 3b 18) (b) (1) (2 3b 18) (b leq 5) (2) (4 3b 18) (b leq 4) (3) (6 3b 18) (b leq 4) (4) (8 3b 18) (b leq 3) (5) (10 3b 18) (b leq 2)

Checking the valid values for each (a):

(a 1, b 4, c 1: 2(1) 3(4) 14 equiv 2 pmod{4}; a^2b^4 16) (a 4, b 2, c 1: 2(4) 3(2) 14 equiv 2 pmod{4}; a^2b^4 256)

The maximum value of (a^2b^4) occurs with (a 1, b 4, c 1) and (a 4, b 2, c 1), both yielding: 256.

Thus, the maximum value of (a^2b^4) is boxed{256}.