Question

Find the number of integral solutions of xyz = 2003​

Answer 1

Answer: hello,

here is your answer:

There's a strong formula, famously known as the stars and bars algorithm, which simplifies this solution for any product.

I'll just detail the solution once I've explained the formula. It's used, ideally, for counting the number of ways of distributing 'n' identical balls into 'r' different boxes. The number of ways of doing that is (n+r-1)C(r-1). The formula can be easily remembered but hardly the applications understood. Let's understand what is the primary difference between distributing 'n' IDENTICAL balls and 'n' DISTINCT balls. The idea behind DISTINCT balls is that in this case, not only how many balls go into a particular box matter, but also WHICH ones matter. But when they are IDENTICAL balls, only thing that matters is how many goes into a specific box.

Here XYZ =24 = 2^3*3

So, we have three identical 2s to distribute into three boxes (X,Y,Z). The number of ways of doing that would be (3+3-1)C(3-1) = 5C2 = 10 ways.

Correspondingly, the number of ways of distributing the 3 into the three boxes must be 3 ways

So, we have a total of 10*3=30 ways of distributing

Step-by-step explanation:

Answer 2

Answer:

Three integral Solutions Possible.

Step-by-step explanation:

xyz = 2003

As we know a Prime number is a number whose factor is 1 and itself the number.

And here, in this case, 2003 is a Prime Number.

So, 2003 = 1 × 1 × 2003

So, x, y, or z can be 1, 1, 2003.

Number of integral solution of x, y, & z are 3.

x = y = 1, z = 2003

x = z = 1, y = 2003

x = 2003, y = z = 1

< hope you got that! >

Thanks!