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Difficulty: Very hardPositive and Negative Number Properties

If xx, yy, and zz are integers satisfying x<y<0<zx < y < 0 < z, x+y+z=0x + y + z = 0, and xyz=160x y z = 160, what is the value of z−xz - x?

Answer: 18

Answer

The value of z−xz - x is 1818.
By defining positive variables a=−xa = -x and b=−yb = -y, the given inequality x<y<0x < y < 0 implies a>b>0a > b > 0. Since x+y+z=0x + y + z = 0, z=a+bz = a + b. The product condition xyz=160xyz = 160 translates to ab(a+b)=160ab(a+b) = 160. Testing positive integer values reveals that b=2b = 2 and a=8a = 8 is the unique solution satisfying a>b>0a > b > 0. This gives x=−8x = -8, y=−2y = -2, and z=10z = 10, making z−x=10−(−8)=18z - x = 10 - (-8) = 18.

Step-by-Step Solution

1
Set up positive variables for the negative integers
Let a=−xa = -x and b=−yb = -y, where aa and bb are positive integers with a>b>0a > b > 0. From x+y+z=0x + y + z = 0, we get −a−b+z=0-a - b + z = 0, so z=a+bz = a + b.
Converting negative integers to positive magnitude variables simplifies sign analysis in products and sums.
2
Substitute variables into the product equation
Substituting x=−ax = -a, y=−by = -b, and z=a+bz = a + b into xyz=160x y z = 160 gives (−a)(−b)(a+b)=160(-a)(-b)(a + b) = 160, which simplifies to ab(a+b)=160a b (a + b) = 160.
Multiplying two negative numbers yields a positive product, simplifying the product expression.
3
Solve for positive integer pairs (a,b)(a, b) with a>ba > b
Testing integer values of bb:
- If b=1b = 1, a(a+1)=160a(a+1) = 160 (no integer solution as 12×13=15612 \times 13 = 156).
- If b=2b = 2, 2a(a+2)=160  ⟹  a(a+2)=80  ⟹  a=82a(a+2) = 160 \implies a(a+2) = 80 \implies a = 8.
- If b=3b = 3, 3a(a+3)=160  ⟹  a(a+3)=53.333a(a+3) = 160 \implies a(a+3) = 53.33 (not an integer).
- If b=4b = 4, 4a(a+4)=160  ⟹  a(a+4)=404a(a+4) = 160 \implies a(a+4) = 40 (no integer solution).
- If b≥5b \ge 5, a>b  ⟹  a≥6a > b \implies a \ge 6, so ab(a+b)≥6×5×11=330>160a b (a+b) \ge 6 \times 5 \times 11 = 330 > 160.
Thus, the only valid integer pair is a=8a = 8 and b=2b = 2.
Systematically checking integer factors under inequality constraints guarantees finding all unique solutions.
4
Calculate the target expression z−xz - x
Since a=8a = 8 and b=2b = 2, we have x=−8x = -8, y=−2y = -2, and z=8+2=10z = 8 + 2 = 10. Therefore, z−x=10−(−8)=18z - x = 10 - (-8) = 18.
Evaluating z−xz - x using the identified values completes the solution.

Key Concept

Positive and Negative Number Properties and Inequality Constraints
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Topics

Positive and Negative Number Properties

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