Question

How many solutions are there to the equation x1 x2 x3 x4 x5 x6 = 29, where xi , i = 1, 2, 3, 4, 5, 6, is a nonnegative integer such that a) xi > 1 for i = 1,2,3,4,5,6? b) x1 ≥1,x2 ≥2,x3 ≥3,x4 ≥4,x5 >5,andx6 ≥6? c) x1 ≤ 5? d) x1 <8andx2 >8?

Answer 1

The number of solutions to the given equation depends on the conditions given in each part. For some cases, there is only one solution, while for others, there can be multiple solutions.

Step-by-step explanation:

To answer this question, we need to determine the number of solutions to the equation x1x2x3x4x5x6 = 29, while considering the given conditions.

a) In this case, we want xi > 1 for i = 1,2,3,4,5,6. We can rearrange the equation as (x1-1)(x2-1)(x3-1)(x4-1)(x5-1)(x6-1) = 29 - 6 = 23. Since 23 is a prime number, there is only one way to factorize it as a product of 6 integers. Therefore, there is 1 solution.

b) In this case, we have specific conditions for the values of xi. We can set up a system of linear equations using the given inequalities and solve for the values of xi. By solving the system, we find that there is 1 solution.

c) In this case, x1 can take on values from 0 to 5, so there are 6 possible solutions.

d) In this case, x1 can be less than 8 and x2 can be greater than 8. However, it does not guarantee a unique solution. Therefore, there can be multiple solutions to this equation.