Question

find number of integerd solutions of

$x \cdot y \cdot z \cdot w \cdot = 2^{10} \cdot 3^5 \cdot 5^8 \cdot 7^9$

Answer 1

4651046400

Answer 2

We are to count the number of 4‐tuples of integers $(x,y,z,w)$ (which may be positive or negative) satisfying

$$x\cdot y\cdot z\cdot w= 2^{10}\cdot 3^5\cdot 5^8\cdot 7^9.$$

Step 1. Count the ways to distribute the prime factors (ignoring signs):

For any prime $p$ with exponent $r$ in the number, we assign its exponent among the four variables by solving

$$a_1+a_2+a_3+a_4 = r, \quad a_i\ge 0.$$

The number of solutions is $\binom{r+4-1}{4-1}=\binom{r+3}{3}$.

• For $2^{10}$: $\binom{10+3}{3}=\binom{13}{3}=286.$
• For $3^{5}$: $\binom{5+3}{3}=\binom{8}{3}=56.$
• For $5^{8}$: $\binom{8+3}{3}=\binom{11}{3}=165.$
• For $7^{9}$: $\binom{9+3}{3}=\binom{12}{3}=220.$

Thus, the total number of ways (for the absolute values) is:

$$286\times 56\times 165\times 220.$$

Step 2. Include the sign distributions:

Since the product $x\cdot y\cdot z\cdot w$ is positive, an even number of the variables must be negative.

For 4 variables:

• 0 negatives: $\binom{4}{0} = 1$ way,
• 2 negatives: $\binom{4}{2} = 6$ ways,
• 4 negatives: $\binom{4}{4} = 1$ way.

So, there are $1+6+1=8$ ways to assign signs.

Step 3. Multiply everything together:

The total number of solutions is:

$$8 \cdot \Big(286\times 56\times 165\times 220\Big).$$

Let’s compute step‐by‐step:

1. $286\times 56$:

  $286\times 50=14300,$

  $286\times 6=1716,$

  Sum = $14300+1716=16016.$

2. $165\times 220$:

  $165\times 200=33000,$

  $165\times 20=3300,$

  Sum = $33000+3300=36300.$

3. Now, $16016\times 36300$:

  First, note that $36300 = 363 \times 100$.

  Calculate $16016\times 363$:

  - $16016\times 300 = 4\,804\,800,$

  - $16016\times 60 = 960\,960,$

  - $16016\times 3 = 48\,048,$

  Sum = $4\,804\,800+960\,960+48\,048 = 5\,813\,808.$

  Thus, $16016\times 36300 = 5\,813\,808\times 100 = 581\,380\,800.$

4. Finally, multiply by 8:

  $581\,380\,800\times 8 = 4\,651\,046\,400.$

Final Answer:

$$\boxed{4651046400}$$