ASVAB · Mathematics Knowledge · 6 min read · Open lesson
Number Properties and Order of Operations
- Understand the classification of integers, absolute value, and the rules governing arithmetic operations and parity.
- Apply divisibility rules and prime factorizations to determine factors, multiples, greatest common factors, and least common multiples.
- Evaluate multi-step numerical expressions correctly using the standard PEMDAS order of operations without a calculator.
Integers, the Number Line, and Absolute Value
The Mathematics Knowledge (MK) subtest covers high school math concepts, including arithmetic, algebra, and geometry. Along with Arithmetic Reasoning (AR), Word Knowledge (WK), and Paragraph Comprehension (PC), your MK score is combined to determine your Armed Forces Qualification Test (AFQT) score. Because scores required for enlistment or specific jobs are set by each military branch and can change over time, building a dependable math foundation is an essential step in your preparation.
A central building block of mathematics is the set of integers. The set of integers consists of positive counting numbers, their negative counterparts, and zero. Integers have no fractional part, although expressions such as 6/2 and 3.0 can represent integers because they evaluate to whole quantities. The integer zero holds a unique position because it is neither positive nor negative.
On a horizontal number line, numbers to the left are smaller than numbers to the right. Consequently, every positive integer is greater than any negative integer, regardless of the magnitude of the negative number. For example, moving toward the right leads to larger values, while moving to the left leads to smaller values.
The absolute value of a number represents its non-negative distance from zero on a number line. Because distance cannot be negative, the absolute value of any real number is always greater than or equal to zero. If a number is positive or zero, its absolute value is the number itself. If a number is negative, its absolute value is the positive opposite of that number. However, if a negative sign is placed outside an absolute value symbol, it makes the final evaluated result negative or zero.
Integer Arithmetic and Parity Rules
Performing arithmetic with signed integers requires careful attention to signs. Adding two positive integers always produces a positive sum, and adding two negative integers always produces a negative sum. To add integers with opposite signs, subtract the smaller absolute value from the larger absolute value. You then assign the sign of the integer with the larger absolute value to the result, or write 0 if the absolute values are equal.
Subtraction is handled by converting the problem into addition: subtracting an integer is mathematically identical to adding its opposite. When working with multiplication and division, the sign rules depend entirely on whether the signs match. Multiplying or dividing two integers with the same sign always produces a positive result, whereas multiplying or dividing two integers with different signs always produces a negative result.
Integers are also categorized by parity as even or odd. An even integer is any integer divisible by 2 with no remainder, while an odd integer is an integer that is not divisible by 2; for example, both 3 and −3 are odd. Parity rules allow you to verify your arithmetic quickly during test-taking.
The sum or difference of two even integers is always even, and the sum or difference of two odd integers is always even. However, the sum or difference of an even integer and an odd integer is always odd. For multiplication, the product of two odd integers is always odd, whereas the product of any integer and an even integer is always even.
| Operation | Combination | Resulting Parity |
|---|---|---|
| Addition or Subtraction | Even and Even | Always Even |
| Addition or Subtraction | Odd and Odd | Always Even |
| Addition or Subtraction | Even and Odd | Always Odd |
| Multiplication | Odd and Odd | Always Odd |
| Multiplication | Any Integer and Even | Always Even |
Factors, Multiples, and Divisibility Rules
An integer factor of an integer is a nonzero integer that divides it with no remainder. For example, both 3 and −3 are factors of 12 because both divide 12 evenly. In contrast, a multiple of an integer is the product of that integer and another integer, which may be positive, negative, or zero.
Because you cannot use a calculator on the MK subtest, divisibility rules help you identify factors quickly. A whole number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). A whole number is divisible by 5 if its last digit is 0 or 5, and it is divisible by 10 if its final digit is 0.
Other divisibility tests involve examining combinations of digits. A whole number is divisible by 4 if the number formed by its last two digits is divisible by 4, and it is divisible by 8 if the number formed by its last three digits is divisible by 8.
Summing the digits offers another shortcut: a whole number is divisible by 3 if the sum of all its digits is divisible by 3, and it is divisible by 9 if the sum of all its digits is divisible by 9. Furthermore, a whole number is divisible by 6 if it is divisible by both 2 and 3.
Prime Numbers, Factorization, GCF, and LCM
Positive integers greater than 1 are classified as either prime or composite. A prime number is a positive integer strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself. A composite number is a positive integer strictly greater than 1 that has more than two distinct positive factors. The number 1 is neither prime nor composite because it has only one positive factor, and the number 0 is also neither prime nor composite.
The number 2 is the smallest prime number and the only even prime number. Memorizing the early primes saves valuable time. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Every composite number has a unique prime factorization, apart from the order in which the prime factors are written.
Prime factorization provides an efficient method for finding the Greatest Common Factor (GCF) and the Least Common Multiple (LCM). For two integers that are not both zero, the Greatest Common Factor (GCF) is the largest positive integer that divides both evenly. For two nonzero integers, factor their absolute values and multiply the lowest powers of their shared prime factors; if they share none, the GCF is 1. If one number is zero, the GCF is the absolute value of the other, for example, the GCF of 0 and 12 is 12.
For two nonzero integers, the Least Common Multiple (LCM) is the smallest positive integer that is a multiple of both. To find the LCM using prime factorization, multiply the highest power of every prime factor present in either factorization. This systematic approach eliminates guesswork and prevents calculation errors.
Order of Operations and Exponents
When an expression contains multiple mathematical operations, you must follow the standard order of operations known by the acronym PEMDAS. This rule requires you to simplify grouping symbols first, followed by exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.
Grouping symbols include parentheses, brackets, braces, fraction bars, and radical signs. When simplifying nested grouping symbols, operations must be performed from the innermost set outward. A horizontal fraction bar acts as a grouping symbol, meaning the numerator and the denominator must each be evaluated fully before dividing.
A common point of confusion involves equal-priority operations. Multiplication and division have equal priority in the order of operations and must be calculated strictly from left to right as they appear. Similarly, addition and subtraction have equal priority in the order of operations and must be calculated from left to right.
Rules governing exponents require strict attention to notation. An exponent applies only to the base immediately preceding it unless grouping symbols enclose a larger expression. For any real number x, (−x)^2 equals x^2 and is nonnegative, while −x^2 means −(x^2) and is nonpositive. In addition, any nonzero number raised to the zero power equals 1.
Common mistakes
- Treating the number 1 or the number 0 as a prime number, when in fact neither 1 nor 0 is prime or composite.
- Assuming multiplication must always precede division, or that addition must always precede subtraction, rather than working each pair from left to right.
- Confusing expressions like (−x)^2 and −x^2; (−x)^2 is nonnegative because the negative sign is inside the parentheses, whereas −x^2 represents −(x^2) and is nonpositive.
- Dividing before fully evaluating the numerator and denominator when an expression contains a horizontal fraction bar.
Summary
- The set of integers contains positive counting numbers, their negative counterparts, and zero (which is neither positive nor negative); numbers to the left on a number line are smaller than numbers to the right.
- Absolute value is the non-negative distance of a number from zero on a number line; a negative sign outside the absolute value makes the final evaluated result negative or zero.
- Adding integers with like signs preserves the sign, while adding integers with opposite signs requires subtracting the smaller absolute value from the larger and keeping the sign of the larger absolute value.
- Divisibility rules allow rapid identification of factors without a calculator, including checking the final digits for 2, 4, 5, 8, and 10, or digit sums for 3 and 9.
- The number 2 is the smallest and only even prime; prime factorization yields the GCF by multiplying the lowest powers of shared prime factors and the LCM by multiplying the highest powers of all prime factors present.
- PEMDAS governs order of operations: innermost grouping symbols first (including horizontal fraction bars), then exponents, followed by multiplication and division from left to right, and addition and subtraction from left to right.
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