If x, y, z are positive integers, such that xy + x + y = 215, yz + y + z = 161 and zx + z + x = 107, then (y – z)/(x – z) =

1. Break Down the Problem

We have three equations involving positive integers $x$, $y$, and $z$:

1. $xy + x + y = 215$
2. $yz + y + z = 161$
3. $zx + z + x = 107$

We need to find the value of $\frac{y - z}{x - z}$.

2. Relevant Concepts

We can rearrange the equations to express them in a more manageable form:

• From $xy + x + y = 215$, we can rearrange to $xy + x + y - 215 = 0$.
• Similar rearrangements can be made for the other two equations.

Let's rewrite each equation:

• $xy + x + y = 215 \implies (x + 1)(y + 1) = 216$
• $yz + y + z = 161 \implies (y + 1)(z + 1) = 162$
• $zx + z + x = 107 \implies (z + 1)(x + 1) = 108$

3. Analysis and Detail

Now we will factor each equation based on the products obtained:

1. First Equation: $(x + 1)(y + 1) = 216$ The factors of 216 are: $1 \times 216, 2 \times 108, 3 \times 72, 4 \times 54, 6 \times 36, 8 \times 27, 9 \times 24, 12 \times 18$

2. Second Equation: $(y + 1)(z + 1) = 162$ The factors of 162 are: $1 \times 162, 2 \times 81, 3 \times 54, 6 \times 27, 9 \times 18, 18 \times 9$

3. Third Equation: $(z + 1)(x + 1) = 108$ The factors of 108 are: $1 \times 108, 2 \times 54, 3 \times 36, 4 \times 27, 6 \times 18, 9 \times 12$

We will find valid combinations $(x + 1, y + 1, z + 1)$ from the factor pairs of each equation.

4. Verification through substitution

Let’s try combining factors that appear among the products:

Starting with the equation $(x + 1)(y + 1) = 216$:

• Let $x + 1 = 9$ and $y + 1 = 24$, then $x = 8$ and $y = 23$.

Next, substituting into $(y + 1)(z + 1) = 162$:

• If $y + 1 = 24$, then $z + 1 = 6.75$ (not valid).

So we discard this pair and try another:

Set $x + 1 = 12$ (then $x = 11$) and $y + 1 = 18$ (then $y = 17$):

• Calculate for $z$ using $(y + 1)(z + 1) = 162$: $18(z + 1) = 162 \implies z + 1 = 9 \implies z = 8$

Now we verify with the third equation:

• Check $(z + 1)(x + 1) = 108$: $9 \times 12 = 108 \quad \text{(valid)}$

Final Answer

Now we can compute: $\frac{y - z}{x - z} = \frac{17 - 8}{11 - 8} = \frac{9}{3} = 3$

Thus, the final answer is: $\boxed{3}$