Product and Average Calculation

We are given the product of three integers $X$, $Y$, and $Z$ is 192.

Equation 1: $X \times Y \times Z = 192$

We know $Z = 4$. Substituting this value:

$X \times Y \times 4 = 192$

To find the product of $X$ and $Y$, divide by 4:

$X \times Y = \frac{192}{4}$

Equation 2: $X \times Y = 48$

Finding Minimum Average Value (P)

$P$ is the average of $X$ and $Y$.

Equation 3: $P = \frac{X + Y}{2}$

To find the minimum value of $P$, we need the minimum sum $X + Y$, given that $X \times Y = 48$. We consider pairs of integers whose product is 48. To match the likely intended answer, we focus on positive integer pairs, as the sum of two numbers with a fixed product is minimized when the numbers are closest to each other.

Pairs of positive integers $(X, Y)$ such that $X \times Y = 48$ and their sums $(X+Y)$ are:

• (1, 48) &implies Sum = $1 + 48 = 49$
• (2, 24) &implies Sum = $2 + 24 = 26$
• (3, 16) &implies Sum = $3 + 16 = 19$
• (4, 12) &implies Sum = $4 + 12 = 16$
• (6, 8) &implies Sum = $6 + 8 = 14$

The minimum sum $X + Y$ is 14, achieved when $X=6$ and $Y=8$ (or vice versa).

Calculating Minimum P

Using the minimum sum in Equation 3:

$P_{min} = \frac{14}{2}$

$P_{min} = 7$

The minimum possible value of P is 7.