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.

To find the minimum possible value of P, we can break down the problem step-by-step:

Step 1: Simplify the product equation
We are given that the product of three integers X, Y, and Z is 192:
$$X·Y·Z=192$$
We are also given that:
$Z=4$
Substituting 4 for Z in the product equation gives:
$$X·Y·4=192$$
Dividing both sides by 4, we find the product of X and Y:
$$X·Y=48$$

Step 2: Express P in terms of X and Y
We are given that P is the average of X and Y:
$$P=\frac{X+Y}{2}$$
To find the minimum possible value of P, we need to find the pair of factors X and Y that minimizes their sum:
$X+Y$

Step 3: Analyze the factor pairs of 48
Since the options provided are positive values, we consider positive integer pairs:
$(X,Y)$
whose product is 48. Let's test all possible positive factor pairs of 48 and calculate their corresponding average P:
• For 1 and 48:
$P=\frac{1+48}{2}=24.5$
• For 2 and 24:
$P=\frac{2+24}{2}=13$
• For 3 and 16:
$P=\frac{3+16}{2}=9.5$
• For 4 and 12:
$P=\frac{4+12}{2}=8$
• For 6 and 8:
$P=\frac{6+8}{2}=7$

Step 4: Determine the minimum value
The minimum average occurs when the two factor values are closest together, which is 6 and 8. Therefore, the minimum possible value of P is 7.