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?
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Understanding the Problem: Minimum Average of Integers
The question asks us to find the minimum possible value of P, where P is the average of two integers X and Y. We are given that the product of three integers X, Y, and Z is 192, and Z is specifically 4.
Breaking Down the Given Information
Let's list out the facts provided:
- The product of three integers X, Y, and Z is 192. Mathematically, this can be written as: \( \text{X} \times \text{Y} \times \text{Z} = 192 \)
- The value of Z is given as 4.
- P is defined as the average of X and Y. This means: \( \text{P} = \frac{\text{X} + \text{Y}}{2} \)
Our goal is to find the smallest possible value for P.
Calculating the Product of X and Y
We know \( \text{X} \times \text{Y} \times \text{Z} = 192 \) and \( \text{Z} = 4 \). We can substitute the value of Z into the first equation:
\( \text{X} \times \text{Y} \times 4 = 192 \)
To find the product of X and Y, we divide both sides by 4:
\( \text{X} \times \text{Y} = \frac{192}{4} \)
\( \text{X} \times \text{Y} = 48 \)
So, we are looking for two integers X and Y whose product is 48. Both X and Y must be integers.
Finding Pairs of Integers (X, Y) for a Product of 48
To minimize the average \( \text{P} = \frac{\text{X} + \text{Y}}{2} \), we need to minimize the sum \( \text{X} + \text{Y} \). For a fixed positive product of two integers, their sum is minimized when the two integers are as close to each other as possible. Since the options provided are all positive, we will consider positive integer pairs for X and Y.
Let's list the positive integer pairs (factors) of 48 and calculate their sum (X+Y) and the average P:
| X | Y | Product (\( \text{X} \times \text{Y} \)) | Sum (\( \text{X} + \text{Y} \)) | Average (\( \text{P} = \frac{\text{X} + \text{Y}}{2} \)) |
|---|---|---|---|---|
| 1 | 48 | 48 | \( 1 + 48 = 49 \) | \( \frac{49}{2} = 24.5 \) |
| 2 | 24 | 48 | \( 2 + 24 = 26 \) | \( \frac{26}{2} = 13 \) |
| 3 | 16 | 48 | \( 3 + 16 = 19 \) | \( \frac{19}{2} = 9.5 \) |
| 4 | 12 | 48 | \( 4 + 12 = 16 \) | \( \frac{16}{2} = 8 \) |
| 6 | 8 | 48 | \( 6 + 8 = 14 \) | \( \frac{14}{2} = 7 \) |
From the table, we can see that as X and Y get closer to each other, their sum decreases. The closest positive integer pair for 48 is (6, 8).
Minimum Possible Value of P
By examining the calculated values of P in the table, the minimum positive value for P occurs when X and Y are 6 and 8 (or 8 and 6), which gives a sum of 14. The average P is then:
\( \text{P} = \frac{6 + 8}{2} = \frac{14}{2} = 7 \)
Therefore, the minimum possible value of P is 7.
Important Questions from Numerical Computation
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