This GRE maths sample question is a Problem Solving question in Number Properties. Concept tested: Least value of the product of three numbers whose range is provided as an inequality of absolute values.

Question 9: x, y, and z are distinct integers such that |x|, |y|, and |z| ≤ 5. What is the least possible value of xyz?

- 0
- -125
- -60
- -100
- -75

From INR 2999

|x| ≤ 5.

which means that x can take values among -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5.

The same holds good for y and z.

The question specifies that the numbers are distinct.

Hence, x ≠ y ≠ z.

We need to find the least value of the product of xyz.

x, y, and z can take both negative and positive values.

Hence, the least value of the product will be when the product is negative.

In negative numbers, larger the magnitude, lower the value.

Hence, we need to select numbers such that the magnitude of the product to be as high as possible and the sign of the product to be negative.

As a first step let us maximize the magnitude keeping in mind the numbers are distinct.

We can take 5 and -5 for two of the numbers and 4 or -4 for the 3rd number.

To ensure that the product is negative, two of these numbers have to be positive and one has to be negative.

The values that x, y, and z can take such that the product is the least is when the values are -5, 5, and 4.

The product, xyz when the integers are -5, 5, and 4 is -100.

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