Which of the following numbers is divisible by both 3 and 4?

A number is divisible by 3 if the sum of its digits is divisible by 3, and divisible by 4 if its last two digits form a number divisible by 4. For 468: the digit sum is 4 + 6 + 8 = 18 (divisible by 3), and the last two digits form 68 (68 ÷ 4 = 17, divisible by 4). Thus, 468 meets both criteria.

What is the prime factorization of 504 written in exponential form?

2³ × 3² × 7. Dividing 504 by prime numbers step-by-step: 504 ÷ 2 = 252, 252 ÷ 2 = 126, 126 ÷ 2 = 63, 63 ÷ 3 = 21, 21 ÷ 3 = 7, and 7 is prime. Collecting the prime factors gives three 2s, two 3s, and one 7, which equals 2³ × 3² × 7.

What is the Greatest Common Factor (GCF) of 84 and 126?

The prime factorization of 84 is 2² × 3 × 7 and the prime factorization of 126 is 2 × 3² × 7. Taking the lowest power of each common prime factor gives 2¹ × 3¹ × 7¹ = 42.

What is the Least Common Multiple (LCM) of 36 and 45?

Prime factorizing both numbers gives 36 = 2² × 3² and 45 = 3² × 5. The LCM takes the highest power of every prime factor present: 2² × 3² × 5 = 4 × 9 × 5 = 180.

Number Properties, Factors & Prime Factoriza | Free Guide 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	$a + b = b + a$	$a \times b = b \times a$	Order of terms can be swapped without changing the result.
Associative	$(a + b) + c = a + (b + c)$	$(a \times b) \times c = a \times (b \times c)$	Grouping of terms can be changed without changing the result.
Distributive	$a(b + c) = ab + ac$	$a(b - c) = ab - ac$	Multiplying a sum is equal to multiplying each addend individually.
Identity	$a + 0 = a$	$a \times 1 = a$	Adding $0$ or multiplying by $1$ leaves the value unchanged.
Zero Property	—	$a \times 0 = 0$	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!

$10 - 4 = 6$ , but $4 - 10 = -6 \implies 10 - 4 \neq 4 - 10$ .
$24 \div 6 = 4$ , but $6 \div 24 = \frac{1}{4} \implies 24 \div 6 \neq 6 \div 24$ .

Prime vs. Composite Numbers

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

Special Numbers: $0$ and $1$

The Number $1$ is neither prime nor composite because it has only one positive factor ( $1$ ).
The Number $0$ is neither prime nor composite.
The Number $2$ is the only even prime number. All other even numbers are composite because they are divisible by $2$ .

Prime Numbers under 50 Reference List

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

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 ( $0, 2, 4, 6, 8$ ).	$4,738$ ends in $8 \implies$ Divisible
3	The sum of all digits is divisible by $3$ .	$468 \to 4+6+8 = 18$ ( $18 \div 3 = 6$ ) $\implies$ Divisible
4	The last two digits form a number divisible by $4$ .	$1,324 \to 24$ ( $24 \div 4 = 6$ ) $\implies$ Divisible
5	The last digit is $0$ or $5$ .	$895$ ends in $5 \implies$ Divisible
6	The number is divisible by both 2 and 3.	$468$ is even AND sum is $18 \implies$ Divisible
9	The sum of all digits is divisible by $9$ .	$7,821 \to 7+8+2+1 = 18$ ( $18 \div 9 = 2$ ) $\implies$ Divisible
10	The last digit is $0$ .	$3,450$ ends in $0 \implies$ Divisible

Prime Factorization & Factor Trees

The Fundamental Theorem of Arithmetic states that every integer greater than $1$ 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 $360$

Split $360$ into any two factors: $36 \times 10$ .
Split $36$ into $6 \times 6$ ; split $10$ into $2 \times 5$ .
Split each $6$ into $2 \times 3$ .
Collect all prime ends: $2, 3, 2, 3, 2, 5$ .
Write using exponents in ascending prime order: $360 = 2^3 \times 3^2 \times 5$

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: $72$ and $108$ .

Express both numbers as prime factorizations: $72 = 2^3 \times 3^2$ $108 = 2^2 \times 3^3$
To find GCF: Take the smallest power of each common prime factor: $\text{GCF}(72, 108) = 2^2 \times 3^2 = 4 \times 9 = \mathbf{36}$
To find LCM: Take the highest power of every prime factor present: $\text{LCM}(72, 108) = 2^3 \times 3^3 = 8 \times 27 = \mathbf{216}$

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

ACCUPLACER Arithmetic

1.3 Number Properties, Factors & Prime Factorization

Key Takeaways

1.3 Number Properties, Factors & Prime Factorization

Fundamental Properties of Whole Numbers

Prime vs. Composite Numbers

Special Numbers: $0$ and $1$

Prime Numbers under 50 Reference List

Divisibility Rules

Prime Factorization & Factor Trees

Factor Tree Method for $360$

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

Method: Finding GCF and LCM via Prime Factorization

ACCUPLACER Exam Traps & Common Errors

ACCUPLACER Arithmetic

1.3 Number Properties, Factors & Prime Factorization

Key Takeaways

1.3 Number Properties, Factors & Prime Factorization

Fundamental Properties of Whole Numbers

Prime vs. Composite Numbers

Special Numbers: 000 and 111

Prime Numbers under 50 Reference List

Divisibility Rules

Prime Factorization & Factor Trees

Factor Tree Method for 360360360

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

Method: Finding GCF and LCM via Prime Factorization

ACCUPLACER Exam Traps & Common Errors

Special Numbers: $0$ and $1$

Factor Tree Method for $360$