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Properties of Integers

11 questions

All types11Multiple choice7True / False2Numeric2
Question 1Question →

If xx and yy are both negative integers, the value of x−yx - y must also be a negative integer.

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Answer: False

Answer

False
The statement is false. When evaluating x−yx - y with negative integers, the operation simplifies to adding a positive number. If the magnitude of yy is larger than the magnitude of xx, the result is positive. For instance, if x=−1x = -1 and y=−5y = -5, then −1−(−5)=4-1 - (-5) = 4.

Step-by-Step Solution

1
Analyze the expression x−yx - y given that both variables represent negative integers.
Since yy is negative, subtracting yy is identical to adding the absolute value of yy. The expression becomes x+∣y∣x + |y|.
Understanding how double negatives simplify is essential for evaluating integer properties.
2
Test the statement using a counterexample where yy is further left on the number line than xx.
Assume x=−3x = -3 and y=−10y = -10. Substitute these integers into the original expression: −3−(−10)-3 - (-10).
A universally phrased statement containing the word 'must' can be entirely disproven with just a single counterexample.
3
Evaluate the chosen counterexample.
−3−(−10)=−3+10=7-3 - (-10) = -3 + 10 = 7. Because 77 is a positive integer, the expression does not always yield a negative result.
The positive outcome proves the initial statement is definitively false.

Key Concept

Integer Subtraction and Sign Rules
Estimated Time:35s
Question 2Question →

If x=−3x = -3 and y=−5y = -5, what is the value of the following expression?

x2+2x×∣y∣−(x−y)x^2 + 2x \times |y| - (x - y)
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Answer: -23

Answer

The correct answer is -23.
By accurately applying the rules for negative integers, (-3)^2 evaluates to a positive 9. The absolute value |-5| evaluates to a positive 5. Following the order of operations, the multiplication 2(-3) * 5 is performed next, yielding -30. Finally, subtracting the evaluation of the parenthetical group (2) from the sum of 9 and -30 results in exactly -23.

Step-by-Step Solution

1
Substitute the values of x and y into the expression.
(−3)2+2(−3)×∣−5∣−(−3−(−5))(-3)^2 + 2(-3) \times |-5| - (-3 - (-5))
Direct substitution is the first step in evaluating algebraic expressions.
2
Evaluate the terms inside parentheses and absolute value brackets.
∣−5∣=5|-5| = 5 and (−3−(−5))=(−3+5)=2(-3 - (-5)) = (-3 + 5) = 2. The expression becomes: (−3)2+2(−3)×5−2(-3)^2 + 2(-3) \times 5 - 2
According to the order of operations, grouping symbols like absolute value and parentheses must be resolved first.
3
Evaluate the exponent.
(−3)2=9(-3)^2 = 9. The expression becomes: 9+2(−3)×5−29 + 2(-3) \times 5 - 2
Exponents must be calculated before multiplication, addition, and subtraction. A negative number squared yields a positive result.
4
Perform the multiplication.
2(−3)×5=−6×5=−302(-3) \times 5 = -6 \times 5 = -30. The expression becomes: 9+(−30)−29 + (-30) - 2
Multiplication is performed before addition and subtraction.
5
Perform the remaining addition and subtraction from left to right.
9−30−2=−21−2=−239 - 30 - 2 = -21 - 2 = -23
Addition and subtraction are the final steps in the order of operations.

Key Concept

Evaluating expressions using properties of negative integers and absolute value
Estimated Time:1m 0s
Question 3Question →

What is the value of the expression −3×∣−2∣−(−5)-3 \times |-2| - (-5)?

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Answer: -1

Answer

The correct answer is -1.
The expression must be solved by first finding the absolute value of −2-2, which is 22. Multiplying −3-3 by 22 gives −6-6. Finally, subtracting −5-5 from −6-6 is the exact same as adding 55 to −6-6, which results in −1-1.

Step-by-Step Solution

1
Evaluate the absolute value expression.
∣−2∣=2|-2| = 2
Absolute value represents the distance of a number from zero, which is always non-negative.
2
Perform the multiplication.
−3×2=−6-3 \times 2 = -6
According to the order of operations (PEMDAS), multiplication must be completed before addition and subtraction. Multiplying a negative integer by a positive integer yields a negative integer.
3
Perform the subtraction.
−6−(−5)=−6+5=−1-6 - (-5) = -6 + 5 = -1
Subtracting a negative integer is mathematically equivalent to adding its positive counterpart.

Key Concept

Integer Sign Rules and Order of Operations
Question 4Question →

The product of two negative integers is always a positive integer.

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Answer: True

Answer

True
The statement is correct because the mathematical rule for multiplying two integers with identical negative signs dictates that their product is positive.

Step-by-Step Solution

1
Recall the rules for multiplying integers with the same sign.
A negative number multiplied by another negative number equals a positive number.
This is a fundamental property of integer multiplication.
2
Apply the rule to a concrete example to verify.
For instance, −3×−4=12-3 \times -4 = 12.
Using concrete integers confirms that the product is indeed a positive integer.

Key Concept

Sign Rules for Integer Operations
Estimated Time:15s
Question 5Question →

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?

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Answer: 5

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.
Question 6Question →
Evaluate the following mathematical expression:
∣−12∣−(−4)×2|-12| - (-4) \times 2
Which of the following is the correct calculated value?
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Answer: 2020

Answer

The correct result of the expression is 2020.
Following the order of operations, first evaluate the absolute value: ∣−12∣=12|-12| = 12. Next, perform the multiplication: −4×2=−8-4 \times 2 = -8. Finally, perform the subtraction: 12−(−8)12 - (-8). Since subtracting a negative number is the same as adding a positive number, the expression becomes 12+8=2012 + 8 = 20.

Step-by-Step Solution

1
Evaluate the absolute value
∣−12∣=12|-12| = 12
Absolute value represents the distance from zero on a number line, so it is always non-negative.
2
Perform the multiplication
−4×2=−8-4 \times 2 = -8
According to the standard order of operations (PEMDAS), multiplication must be performed before subtraction.
3
Subtract the product from the absolute value result
12−(−8)=12+8=2012 - (-8) = 12 + 8 = 20
Subtracting a negative integer is mathematically equivalent to adding its positive counterpart.

Key Concept

Applying the order of operations (PEMDAS), evaluating absolute values, and following integer sign rules.
Question 7Question →

What is the absolute difference between the smallest positive integer and the largest negative integer?

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Answer: 2

Answer

2
The smallest positive integer is 1 and the largest negative integer is -1. The absolute difference between them is found by subtracting the two values and taking the absolute value: ∣1−(−1)∣=∣1+1∣=2|1 - (-1)| = |1 + 1| = 2.

Step-by-Step Solution

1
Identify the smallest positive integer.
1
The integer 0 is neither positive nor negative. The positive integers start at 1, making it the smallest positive integer.
2
Identify the largest negative integer.
-1
Negative integers increase in value as they get closer to zero. Therefore, -1 is the largest negative integer.
3
Calculate the absolute difference.
2
The absolute difference represents the total distance between the two numbers on the number line. Calculate this as ∣1−(−1)∣=∣1+1∣=2|1 - (-1)| = |1 + 1| = 2.

Key Concept

Properties of Integers (Number Line and Absolute Value)
Question 8Question →

Let xx be a negative even integer and yy be a negative odd integer. Which of the following expressions will always result in a positive odd integer?

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Answer: x×y+1x \times y + 1

Answer

x×y+1x \times y + 1
When multiplying two negative integers, the result is positive. When multiplying an even integer by an odd integer, the result is always even. Therefore, the product of xx and yy is a positive even integer. Adding 1 to any even integer results in an odd integer. Thus, the expression always yields a positive odd integer.

Step-by-Step Solution

1
Determine the sign and parity of the product x×yx \times y.
x×yx \times y is a positive even integer.
Multiplying two negative numbers yields a positive number. Multiplying an even number by any integer yields an even number.
2
Evaluate the result of adding 1 to the product.
x×y+1x \times y + 1 is a positive odd integer.
Adding 1 to any even integer changes its parity to odd. Adding 1 to a positive number keeps it positive.
3
Check the validity of the other provided expressions.
Only x×y+1x \times y + 1 guarantees a positive odd result.
Other expressions result in negative odd values, positive even values, or values that can fluctuate in sign based on specific magnitudes.

Key Concept

Parity and Sign Properties of Integers
Question 9Question →

The sum of three consecutive negative even integers is −102-102. What is the absolute value of the smallest of these integers?

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Answer: 36

Answer

36
By defining the consecutive even integers algebraically, the smallest integer is correctly identified as −36-36. Taking the absolute value of −36-36 yields 3636.

Step-by-Step Solution

1
Define variables for the three consecutive even integers.
Let the smallest integer be xx. The next consecutive even integers are x+2x + 2 and x+4x + 4.
Consecutive even integers are always 2 units apart.
2
Set up the equation.
x+(x+2)+(x+4)=−102x + (x + 2) + (x + 4) = -102
The sum of the three integers is given as −102-102.
3
Solve for xx.
3x+6=−1023x + 6 = -102 leads to 3x=−1083x = -108, yielding x=−36x = -36.
Isolating xx provides the value of the smallest integer.
4
Calculate the absolute value.
∣−36∣=36|-36| = 36
The question specifically asks for the absolute value of the smallest integer.

Key Concept

Consecutive integers and absolute value properties
Question 10Question →

Which of the following statements about integers is always true?

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Answer: The difference between any integer and its absolute value is always an even integer.

Answer

The statement claiming that the difference between any integer and its absolute value is always an even integer.
For any non-negative integer, the difference between the integer and its absolute value is 0, which is an even integer. For any negative integer, the difference evaluates to twice the integer (2x), which is a multiple of 2 and therefore also even. Thus, the statement holds true for all integers.

Step-by-Step Solution

1
Analyze the behavior of the expression (x - |x|) for non-negative integers.
If x is 0 or positive, |x| = x. Therefore, x - |x| = x - x = 0.
To test the property for the first distinct category of integers.
2
Analyze the behavior of the expression (x - |x|) for negative integers.
If x is negative, |x| = -x. Therefore, x - |x| = x - (-x) = x + x = 2x.
To test the property for the remaining category of integers.
3
Determine the parity (odd/even status) of the results from the previous steps.
The number 0 is an even integer. The expression 2x is inherently a multiple of 2, making it an even integer regardless of the value of x.
To conclude whether the difference is universally even.

Key Concept

Parity and Absolute Value Properties
Question 11Question →

A student is required to identify a calculation that yields a value greater than zero. Which of the following arithmetic expressions produces a positive integer result?

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Answer: (−3)×(−4)−10(-3) \times (-4) - 10

Answer

The expression (−3)×(−4)−10(-3) \times (-4) - 10 produces a positive integer.
The expression (−3)×(−4)−10(-3) \times (-4) - 10 evaluates to a positive integer because the product of −3-3 and −4-4 is a positive 1212, and subtracting 1010 from 1212 leaves 22.

Step-by-Step Solution

1
Evaluate the first expression: −5+(−2)-5 + (-2).
−5+(−2)=−7-5 + (-2) = -7.
Adding a negative integer to a negative integer results in a smaller negative integer.
2
Evaluate the second expression: 4−3×24 - 3 \times 2.
4−6=−24 - 6 = -2.
Following the order of operations (PEMDAS), multiplication is performed before subtraction.
3
Evaluate the third expression: −∣−8∣+5-|-8| + 5.
−8+5=−3-8 + 5 = -3.
The absolute value of −8-8 is positive 88. The negative sign outside makes it −8-8. Adding 55 results in a negative integer.
4
Evaluate the fourth expression: (−3)×(−4)−10(-3) \times (-4) - 10.
12−10=212 - 10 = 2.
Multiplying two negative integers yields a positive integer (1212). Subtracting 1010 from 1212 leaves a positive integer.

Key Concept

Integer operations combining sign rules, absolute value, and the order of operations.
Estimated Time:50s
Properties of Integers Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) | Examkin