The product of three integers X, Y and Z is 192. Z is equal to 4 and P is equal to the average of X and Y. What is the minimum possible value of P?

The correct option is 7.

Let's break down the mathematical derivation step-by-step to understand why this is the correct answer.

We are given that $X$, $Y$, and $Z$ are integers.
The product of these three integers is given as:

$X\cdot Y\cdot Z=192$

We are also given that:

$Z=4$

Substituting the value of $Z$ into the product equation, we get:

$X\cdot Y\cdot 4=192$

Dividing both sides by 4, we find the product of $X$ and $Y$:

$X\cdot Y=\frac{192}{4}=48$

The variable $P$ is defined as the average of the two integers $X$ and $Y$:

$P=\frac{X+Y}{2}$

To find the minimum possible value of $P$, we need to minimize the sum $X+Y$ under the condition that $X\cdot Y=48$ and both $X$ and $Y$ are integers.
Since $X$ and $Y$ can be negative integers, let's look at the possible integer pairs $(X,Y)$ whose product is 48:
If $X$ and $Y$ are positive:

• $X=1$, $Y=48$ ⇒ $X+Y=49$
• $X=2$, $Y=24$ ⇒ $X+Y=26$
• $X=3$, $Y=16$ ⇒ $X+Y=19$
• $X=4$, $Y=12$ ⇒ $X+Y=16$
• $X=6$, $Y=8$ ⇒ $X+Y=14$

• $X=-1$, $Y=-48$ ⇒ $X+Y=-49$
• $X=-2$, $Y=-24$ ⇒ $X+Y=-26$
• $X=-3$, $Y=-16$ ⇒ $X+Y=-19$
• $X=-4$, $Y=-12$ ⇒ $X+Y=-16$
• $X=-6$, $Y=-8$ ⇒ $X+Y=-14$

Generally, the minimum sum $X+Y$ of two integers with a positive product $48$ occurs when both integers are negative and as close to zero as possible (i.e., when they are closest to each other in absolute value). This gives a minimum sum of $-49$, which would result in $P=-24.5$.
However, looking at the given options: 7, 6, 8, and 9.5, all are positive values. This indicates that the question restricts $X$ and $Y$ to be positive integers.

For positive integers $X$ and $Y$, the minimum value of the sum $X+Y$ is achieved when the two numbers are as close to each other as possible. From our list of positive integer factor pairs:

The closest pair is $X=6$ and $Y=8$ (or vice versa), which gives the minimum positive sum:

${(X+Y)}_{\min }=6+8=14$

Using this minimum sum, we calculate the minimum value of the average $P$:

$P_{\min }=\frac{14}{2}=7$

Therefore, the minimum possible value of $P$ is indeed 7.