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Difficulty: HardProperties of Integers

A custom mathematical operation is defined for all integers such that a⋆b=∣a−b∣−(a×b)a \star b = |a - b| - (a \times b). If aa represents the largest negative integer and bb represents the smallest prime number, what is the value of the expression a⋆ba \star b?

  1. 5Answer
  2. B
    1
  3. C
    -1
  4. D
    8

Answer

5
The correct value is 5. First, identify the specific integers based on their definitions: the largest negative integer is -1 (so a = -1), and the smallest prime number is 2 (so b = 2). Substituting these values into the given operation yields |-1 - 2| - (-1 \times 2). Evaluate the grouping inside the absolute value first: |-1 - 2| = |-3| = 3. Next, evaluate the multiplication: -1 \times 2 = -2. Finally, subtract the product from the absolute value result: 3 - (-2), which simplifies to 3 + 2 = 5.

Step-by-Step Solution

1
Identify the numerical values of the integers a and b based on their mathematical properties.
a = -1 and b = 2
The largest negative integer is -1 (as it is closest to zero on the negative side), and the smallest prime number is 2 (the only even prime).
2
Substitute the values of a and b into the custom operation expression.
|-1 - 2| - (-1 \times 2)
Direct substitution is required to evaluate a \star b.
3
Simplify the arithmetic inside the absolute value brackets.
∣−3∣−(−1×2)|-3| - (-1 \times 2)
According to the order of operations, operations inside grouping symbols must be completed first.
4
Evaluate the absolute value and perform the multiplication.
3 - (-2)
The absolute value of -3 is 3 (distance from zero is always positive), and multiplying -1 by 2 yields -2.
5
Perform the final subtraction operation.
5
Subtracting a negative number is mathematically equivalent to adding a positive number: 3 + 2 = 5.

Key Concept

Applying fundamental integer definitions (prime numbers, negative bounds) alongside absolute value rules and the order of operations.
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Topics

Properties of Integers

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