We are given that the product of three integers X, Y, and Z is 192.
$ X \times Y \times Z = 192 $
We know that Z = 4. Substituting this value:
$ X \times Y \times 4 = 192 $
Divide both sides by 4 to find the product of X and Y:
$ X \times Y = \frac{192}{4} $
$ X \times Y = 48 $
We are also 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 minimum possible value of the sum $(X + Y)$, given that $(X \times Y = 48)$.
Since the options provided are positive values, we consider positive integer pairs for X and Y whose product is 48. We look for the pair with the smallest sum.
Finding Minimum Sum (X + Y)
List the pairs of positive integers (X, Y) whose product is 48 and calculate their sum:
- (1, 48): Sum = $(1 + 48) = 49$
- (2, 24): Sum = $(2 + 24) = 26$
- (3, 16): Sum = $(3 + 16) = 19$
- (4, 12): Sum = $(4 + 12) = 16$
- (6, 8): Sum = $(6 + 8) = 14$
The minimum sum $(X + Y)$ among these pairs is 14, which occurs when X=6 and Y=8 (or vice versa).
Calculating Minimum P
Now, use the minimum sum to calculate the minimum value of P:
$ P_{min} = \frac{min(X + Y)}{2} $
$ P_{min} = \frac{14}{2} $
$ P_{min} = 7 $
Therefore, the minimum possible value of P is 7.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
