1.3 Number Properties, Factors & Prime Factorization

ACCUPLACER Arithmetic Exam

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1 Chapter 1: Whole Number Operations

1.1 Basic Whole Number Arithmetic 1.2 Order of Operations & Grouping Symbols 1.3 Number Properties, Factors & Prime Factorization 1.4 Whole Number Estimation, Rounding & Applied Word Problems

2 Chapter 2: Fraction Operations

2.1 Fraction Fundamentals & Simplifying 2.2 Addition & Subtraction of Fractions and Mixed Numbers 2.3 Multiplication & Division of Fractions and Mixed Numbers 2.4 Real-World Fraction Applications & Word Problems

3 Chapter 3: Decimal Operations

3.1 Decimal Place Value, Rounding & Comparing Decimals 3.2 Addition & Subtraction of Decimals 3.3 Multiplication & Division of Decimals 3.4 Decimal Word Problems & Financial Calculations

4 Chapter 4: Percent

4.1 Percent Concepts & Conversions 4.2 Percent of a Quantity & Percent Equations 4.3 Percent Increase, Decrease & Percent Change 4.4 Practical Percent Applications

5 Chapter 5: Number Comparisons & Equivalents

5.1 Ordering Mixed Numerical Forms 5.2 Equivalent Numerical Representations & Conversion Strategies 5.3 Ratios, Unit Rates & Proportional Reasoning

1.3 Number Properties, Factors & Prime Factorization

Key Takeaways Explain This Topic AI

Commutative and Associative properties apply strictly to Addition and Multiplication, never to Subtraction or Division.

A prime number has exactly two positive factors (1 and itself); 1 is neither prime nor composite, and 2 is the only even prime.

Divisibility rules allow quick factor testing: sum of digits for 3 and 9, last two digits for 4, and combining 2 and 3 for 6.

Every whole number greater than 1 has a unique prime factorization expressed as a product of prime powers.

GCF uses the lowest powers of common prime factors, while LCM uses the highest powers of all prime factors present.

Last updated: August 2026

1.3 Number Properties, Factors & Prime Factorization

A thorough understanding of number properties, prime numbers, divisibility rules, and factor relationships allows you to simplify complex arithmetic expressions quickly and accurately on the ACCUPLACER test.

Fundamental Properties of Whole Numbers

Whole numbers follow specific structural rules known as arithmetic properties. These properties allow you to rearrange and regroup terms to make mental arithmetic significantly easier.

Property Addition Formula Multiplication Formula Key Conceptual Principle

Commutative a + b = b + a a + b = b + a a + b = b + a a × b = b × a a \times b = b \times a a × b = b × a Order of terms can be swapped without changing the result.

Associative ( a + b ) + c = a + ( b + c ) (a + b) + c = a + (b + c) ( a + b ) + c = a + ( b + c ) ( a × b ) × c = a × ( b × c ) (a \times b) \times c = a \times (b \times c) ( a × b ) × c = a × ( b × c ) Grouping of terms can be changed without changing the result.

Distributive a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c a ( b − c ) = a b − a c a(b - c) = ab - ac a ( b − c ) = ab − a c Multiplying a sum is equal to multiplying each addend individually.

Identity a + 0 = a a + 0 = a a + 0 = a a × 1 = a a \times 1 = a a × 1 = a Adding 0 0 0 or multiplying by 1 1 1 leaves the value unchanged.

Zero Property — a × 0 = 0 a \times 0 = 0 a × 0 = 0 Any number multiplied by zero equals zero.

Important

Non-Applicability Warning: The Commutative and Associative properties apply ONLY to Addition and Multiplication . They do NOT apply to Subtraction or Division!

10 − 4 = 6 10 - 4 = 6 10 − 4 = 6 , but 4 − 10 = − 6 ⟹ 10 − 4 ≠ 4 − 10 4 - 10 = -6 \implies 10 - 4 \neq 4 - 10 4 − 10 = − 6 ⟹ 10 − 4  = 4 − 10 .

24 ÷ 6 = 4 24 \div 6 = 4 24 ÷ 6 = 4 , but 6 ÷ 24 = 1 4 ⟹ 24 ÷ 6 ≠ 6 ÷ 24 6 \div 24 = \frac{1}{4} \implies 24 \div 6 \neq 6 \div 24 6 ÷ 24 = 4 1 ⟹ 24 ÷ 6  = 6 ÷ 24 .

Prime vs. Composite Numbers

Prime Number: A whole number strictly greater than 1 1 1 that has exactly two distinct positive factors : 1 1 1 and itself.

Composite Number: A whole number strictly greater than 1 1 1 that has more than two positive factors .

Special Numbers: 0 0 0 and 1 1 1

The Number 1 1 1 is neither prime nor composite because it has only one positive factor ( 1 1 1 ).

The Number 0 0 0 is neither prime nor composite.

The Number 2 2 2 is the only even prime number . All other even numbers are composite because they are divisible by 2 2 2 .

Prime Numbers under 50 Reference List

{ 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 } \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\} { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 }

Divisibility Rules

Divisibility rules allow you to determine whether one whole number divides into another without leaving a remainder, without performing full long division.

Divisor Divisibility Condition Rule Practical Example

2 The last digit is even ( 0 , 2 , 4 , 6 , 8 0, 2, 4, 6, 8 0 , 2 , 4 , 6 , 8 ). 4 , 738 4,738 4 , 738 ends in 8 ⟹ 8 \implies 8 ⟹ Divisible

3 The sum of all digits is divisible by 3 3 3 . 468 → 4 + 6 + 8 = 18 468 \to 4+6+8 = 18 468 → 4 + 6 + 8 = 18 ( 18 ÷ 3 = 6 18 \div 3 = 6 18 ÷ 3 = 6 ) ⟹ \implies ⟹ Divisible

4 The last two digits form a number divisible by 4 4 4 . 1 , 324 → 24 1,324 \to 24 1 , 324 → 24 ( 24 ÷ 4 = 6 24 \div 4 = 6 24 ÷ 4 = 6 ) ⟹ \implies ⟹ Divisible

5 The last digit is 0 0 0 or 5 5 5 . 895 895 895 ends in 5 ⟹ 5 \implies 5 ⟹ Divisible
6 The number is divisible by both 2 and 3 . 468 468 468 is even AND sum is 18 ⟹ 18 \implies 18 ⟹ Divisible

9 The sum of all digits is divisible by 9 9 9 . 7 , 821 → 7 + 8 + 2 + 1 = 18 7,821 \to 7+8+2+1 = 18 7 , 821 → 7 + 8 + 2 + 1 = 18 ( 18 ÷ 9 = 2 18 \div 9 = 2 18 ÷ 9 = 2 ) ⟹ \implies ⟹ Divisible

10 The last digit is 0 0 0 . 3 , 450 3,450 3 , 450 ends in 0 ⟹ 0 \implies 0 ⟹ Divisible
Prime Factorization & Factor Trees

The Fundamental Theorem of Arithmetic states that every integer greater than 1 1 1 is either a prime number itself or can be represented as a unique product of prime numbers, up to the order of the factors.

Factor Tree Method for 360 360 360

Split 360 360 360 into any two factors: 36 × 10 36 \times 10 36 × 10 .

Split 36 36 36 into 6 × 6 6 \times 6 6 × 6 ; split 10 10 10 into 2 × 5 2 \times 5 2 × 5 .

Split each 6 6 6 into 2 × 3 2 \times 3 2 × 3 .

Collect all prime ends: 2 , 3 , 2 , 3 , 2 , 5 2, 3, 2, 3, 2, 5 2 , 3 , 2 , 3 , 2 , 5 .

Write using exponents in ascending prime order: 360 = 2 3 × 3 2 × 5 360 = 2^3 \times 3^2 \times 5 360 = 2 3 × 3 2 × 5

Greatest Common Factor (GCF) & Least Common Multiple (LCM)

Greatest Common Factor (GCF): The largest whole number that divides evenly into two or more given numbers without a remainder.

Least Common Multiple (LCM): The smallest non-zero whole number that is a multiple of two or more given numbers.

Method: Finding GCF and LCM via Prime Factorization

Numbers: 72 72 72 and 108 108 108 .

Express both numbers as prime factorizations: 72 = 2 3 × 3 2 72 = 2^3 \times 3^2 72 = 2 3 × 3 2 108 = 2 2 × 3 3 108 = 2^2 \times 3^3 108 = 2 2 × 3 3

To find GCF: Take the smallest power of each common prime factor: GCF ( 72 , 108 ) = 2 2 × 3 2 = 4 × 9 = 36 \text{GCF}(72, 108) = 2^2 \times 3^2 = 4 \times 9 = \mathbf{36} GCF ( 72 , 108 ) = 2 2 × 3 2 = 4 × 9 = 36

To find LCM: Take the highest power of every prime factor present: LCM ( 72 , 108 ) = 2 3 × 3 3 = 8 × 27 = 216 \text{LCM}(72, 108) = 2^3 \times 3^3 = 8 \times 27 = \mathbf{216} LCM ( 72 , 108 ) = 2 3 × 3 3 = 8 × 27 = 216

ACCUPLACER Exam Traps & Common Errors

Caution

Trap 1: Confusing GCF and LCM GCF produces a number less than or equal to the smallest original number. LCM produces a number greater than or equal to the largest original number. Remember: Factors are smaller (they divide into a number); Multiples are larger (a number multiplies out to them).

Warning

Trap 2: Assuming Odd Numbers Are Always Prime Numbers like 9 , 15 , 21 , 25 , 27 , 33 , 35 , 39 , 49 , 51 , 57 , 91 9, 15, 21, 25, 27, 33, 35, 39, 49, 51, 57, 91 9 , 15 , 21 , 25 , 27 , 33 , 35 , 39 , 49 , 51 , 57 , 91 are odd, but they are composite because they have factors other than 1 1 1 and themselves (e.g., 51 = 3 × 17 51 = 3 \times 17 51 = 3 × 17 , 91 = 7 × 13 91 = 7 \times 13 91 = 7 × 13 ). Always test odd numbers against divisibility by 3 , 5 , 7 3, 5, 7 3 , 5 , 7 .