The total number of positive integral solutions (x, y, z) such that xyz = 24 is (A) 36 (B) 45 (C) 24

Q: Wait, why do we use prime factorization? Can't we just list the factors?

Listing factors is fine for small numbers, but it's a trap for larger ones! Prime factorization transforms a complex multiplicative problem into a simple additive one. By focusing on the exponents, we unlock the power of combinatorics.

Q: Why does the Stars and Bars method apply here?

Stars and Bars is the perfect tool for distributing identical items (in this case, the prime factors) into distinct bins (the variables $x, y,$ and $z$). Since the exponents must sum to the total power of the prime in $24$, it maps perfectly to the equation $a_1 + a_2 + a_3 = n$.

Q: Does the order of $x, y, z$ matter?

Yes, absolutely! The problem asks for the number of solutions $(x, y, z)$. This means $(1, 2, 12)$ is considered distinct from $(2, 1, 12)$. The Stars and Bars method inherently accounts for this order because it treats the variables as distinct 'bins'.

Question

The total number of positive integral solutions $(x, y, z)$ such that $xyz = 24$ is

Options:

(A) 36

(B) 45

(C) 24

(D) 30

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler in the world of mathematics! Today, we are not just solving a problem; we are embarking on a journey to uncover the hidden structure of numbers.

We are tasked with finding the number of positive integral solutions $(x, y, z)$ such that $xyz = 24$.

Imagine $24$ as a treasure chest, and we need to distribute its contents into three distinct boxes labeled $x, y,$ and $z$. Let us find out how many ways we can do this.

### The DNA of 24

Whenever we deal with products, the most powerful tool in our arsenal is prime factorization. Think of prime numbers as the fundamental building blocks of all integers.

If we break $24$ down into its prime DNA, we get:

$$24 = 8 	imes 3 = 2^3 	imes 3^1$$

Any positive integer $x, y,$ or $z$ that divides $24$ must be composed of these same prime factors. We can express our variables as:

$$x = 2^{a_1} 3^{b_1}, \quad y = 2^{a_2} 3^{b_2}, \quad z = 2^{a_3} 3^{b_3}$$

Here, the exponents $a_i$ and $b_i$ are non-negative integers. This transformation converts a multiplication problem into an addition problem.

### The Exponent Game

When we multiply $x, y,$ and $z$, the laws of exponents dictate that we add the powers of the same base. For the prime base $2$, the total power must match the power in $24$, which is $3$.

Thus, we get the equation:

$$a_1 + a_2 + a_3 = 3$$

Similarly, for the prime base $3$, the total power must be $1$, leading to:

$$b_1 + b_2 + b_3 = 1$$

Now, we are no longer dealing with $24$; we are dealing with the distribution of exponents. This is where the **Stars and Bars** method shines.

### The Combinatorial Magic

The Stars and Bars method states that the number of ways to distribute $n$ identical items into $r$ distinct bins is given by the formula:

$$\binom{n+r-1}{r-1}$$

For our first equation, $a_1 + a_2 + a_3 = 3$, we have $n=3$ and $r=3$. Plugging these into our formula:

$$\binom{3+3-1}{3-1} = \binom{5}{2} = \frac{5 	imes 4}{2 	imes 1} = 10$$

There are $10$ ways to distribute the powers of $2$. For the second equation, $b_1 + b_2 + b_3 = 1$, we have $n=1$ and $r=3$:

$$\binom{1+3-1}{3-1} = \binom{3}{2} = 3$$

There are $3$ ways to distribute the powers of $3$.

### The Grand Finale

Since the distribution of the powers of $2$ and $3$ are independent, we use the fundamental principle of counting to multiply these possibilities together.

The total number of solutions is:

$$10 	imes 3 = 30$$

We have navigated the complexities of prime factorization and combinatorial distribution to arrive at the final answer of **30**. Remember, every time you face a problem like this, look for the underlying structure.

Animated Solution for Mathematics - Permutations and Combinations: The total number of positive integral solutions $(x, y, z)$ such that $x y z = 24$ is

Select Answer:

Visualized Solution

The Equation

Prime Factorization

Form of Variables

Equating Exponents of $2$

Equating Exponents of $3$

The Stars and Bars Method

Solutions for Exponents of $2$

Solutions for Exponents of $3$

Total Number of Solutions

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The DNA of 24

The Exponent Game

The Combinatorial Magic

The Grand Finale

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Animated Solution for Mathematics - Permutations and Combinations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The total number of positive integral solutions (x,y,z) such that xyz=24 is