Divisibility, prime numbers, prime factorization, greatest common factor, and least common multiple — the hidden structure of whole numbers that supports every fraction, ratio, and algebraic expression
Chapter 1 introduced you to numbers, the number line, and signed operations. You learned how to add, subtract, multiply, and divide with positive and negative values. Chapter 2 goes deeper into the structure of whole numbers themselves.
Numbers are not just values to add or subtract. They have internal structure. Some numbers divide evenly into others. Some numbers are products of smaller numbers multiplied together. Some numbers share common building blocks with other numbers. Understanding this structure is not just an interesting observation. It is a practical tool that will help you simplify fractions, find common denominators, work with ratios, and begin algebraic reasoning.
This chapter teaches you to look at a whole number and ask: "What is this number made of?" The answer to that question will support your work throughout the rest of this textbook and well into Algebra I.
A catering company is preparing trays of food for two events happening at the same time. One event needs trays that each hold 8 sandwiches. Another event needs trays that each hold 12 sandwiches. The kitchen staff wants to make the same number of sandwiches for each tray style without having any sandwiches left over. What is the smallest number of sandwiches they could prepare so that all trays for both events are exactly full?
You may not know how to answer this question yet, but by the end of this chapter you will recognize it immediately as a Least Common Multiple problem and solve it with confidence.
Every time you simplify a fraction, you are using the idea of common factors. Every time you find a common denominator, you are using the idea of common multiples. Every time you factor an algebraic expression in Algebra I, you are extending the skills built here. Divisibility, prime factorization, GCF, and LCM are not isolated topics. They are tools for understanding how numbers fit together, and that understanding will carry you through fractions, ratios, equations, and beyond.
When you look at the number 12, you might first think of it as simply "twelve," the value between eleven and thirteen. But 12 has a richer internal structure. It can be broken apart as 2 × 6, as 3 × 4, as 2 × 2 × 3, or as 1 × 12. It is divisible by 1, 2, 3, 4, 6, and 12. The number 12 appears as the twelfth multiple of 1, the sixth multiple of 2, the fourth multiple of 3, and the third multiple of 4.
This internal structure is not accidental or merely interesting. It has consequences every time you work with 12 in a mathematical context. If you are simplifying the fraction 8/12, you need to know what 8 and 12 share. If you are adding 1/4 + 1/6 and need a common denominator, you need to find the smallest number that both 4 and 6 divide into evenly.
The arithmetic operations from Chapter 1 tell you how to combine numbers. The tools in this chapter tell you how numbers are built. Both perspectives are necessary for fluent mathematics.
Another way to think about number structure is to ask: what pairs or groups of whole numbers multiply together to produce a given number? For 18, you can write: 18 = 2 × 9, or 18 = 3 × 6, or 18 = 1 × 18, or 18 = 2 × 3 × 3. Each of these is a factored form of 18. The numbers 1, 2, 3, 6, 9, and 18 are the factors of 18. Understanding the factors of 18 allows you to simplify fractions with 18 in the numerator or denominator, find common denominators involving 18, and apply ratio and proportion reasoning.
A whole number a is divisible by a whole number b (where b ≠ 0) if dividing a by b produces a whole number quotient with no remainder. If 28 is divisible by 4, then there exists a whole number (7) such that 4 × 7 = 28. Every divisibility statement corresponds to a multiplication statement.
a is divisible by b if a ÷ b produces a whole number with remainder 0.
Equivalently: a is divisible by b if there exists a whole number k such that b × k = a.
| Statement | Division Check | Remainder | Divisible? |
|---|---|---|---|
| 24 divisible by 6? | 24 ÷ 6 = 4 | 0 | Yes |
| 24 divisible by 7? | 24 ÷ 7 ≈ 3.43 | 3 | No |
| 100 divisible by 10? | 100 ÷ 10 = 10 | 0 | Yes |
| 100 divisible by 9? | 100 ÷ 9 ≈ 11.11 | 1 | No |
84 ÷ 6 = 14 remainder 0 → Yes, divisible by 6
84 ÷ 7 = 12 remainder 0 → Yes, divisible by 7
84 ÷ 11 = 7 remainder 7 → No, not divisible by 11
Divisibility is an all-or-nothing property. A number is either divisible by another number or it is not. There is no such thing as "almost divisible."
Performing long division every time you want to test divisibility is slow. Mathematicians have identified patterns, called divisibility rules, that allow you to test whether a number is divisible by certain values using only the digits of the number itself. These rules are especially useful when working with large numbers, checking whether fractions can be simplified, and identifying factors quickly.
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 | 348 ends in 8 → divisible |
| 3 | Sum of digits is divisible by 3 | 471: 4+7+1=12, 12÷3=4 → divisible |
| 4 | Last two digits form a number divisible by 4 | 1,736: 36÷4=9 → divisible |
| 5 | Last digit is 0 or 5 | 345 ends in 5 → divisible |
| 6 | Divisible by both 2 and 3 | 312: even ✓, digit sum 6 ✓ → divisible |
| 9 | Sum of digits is divisible by 9 | 7,254: 7+2+5+4=18, 18÷9=2 → divisible |
| 10 | Last digit is 0 | 560 ends in 0 → divisible |
The rule for 9 is stricter than the rule for 3. Every number divisible by 9 is also divisible by 3, but not every number divisible by 3 is divisible by 9. For example, 12 is divisible by 3 but not by 9.
By 2: Last digit is 0 (even) → Divisible ✓
By 3: Digit sum: 4+3+2+0 = 9, 9÷3 = 3 → Divisible ✓
By 4: Last two digits: 20, 20÷4 = 5 → Divisible ✓
By 5: Last digit is 0 → Divisible ✓
By 6: Divisible by 2 and 3 → Divisible ✓
By 9: Digit sum 9, 9÷9 = 1 → Divisible ✓
By 10: Last digit is 0 → Divisible ✓
Students sometimes apply the rule for 9 (digit sum divisible by 9) when they meant to check for 3, or vice versa. Be deliberate about which rule applies to which number.
Test each number for divisibility by 2, 3, 4, 5, 6, 9, and 10. State which rules apply.
The multiples of a whole number are the products you get when you multiply that number by 1, 2, 3, 4, 5, and so on. Multiples extend outward from a number indefinitely. The list of multiples of any positive whole number is infinite.
If 24 is a multiple of 6, then 24 is also divisible by 6. These two statements say the same thing in different directions: "24 is a multiple of 6" means 6 × 4 = 24, while "24 is divisible by 6" means 24 ÷ 6 = 4 exactly. Understanding that multiples and divisibility are two perspectives on the same relationship will save you effort throughout this chapter.
| Number | First Eight Multiples |
|---|---|
| 4 | 4, 8, 12, 16, 20, 24, 28, 32 |
| 7 | 7, 14, 21, 28, 35, 42, 49, 56 |
| 9 | 9, 18, 27, 36, 45, 54, 63, 72 |
| 12 | 12, 24, 36, 48, 60, 72, 84, 96 |
78 ÷ 6 = 13 (exactly)The factors of a whole number are the whole numbers that divide into it evenly. Unlike multiples, which go on indefinitely, a number has only a finite list of factors. To find all factors of a number, test each whole number beginning with 1 and working upward, listing factor pairs — pairs of numbers that multiply to give the original number — until the pairs begin to repeat.
A practical shortcut: you can stop testing when the candidate factor exceeds the square root of the number. For 36, √36 = 6, so testing 1 through 6 is enough to find all factor pairs.
Factors are smaller than or equal to the original number (for whole numbers greater than 1), and the list is finite. Multiples are greater than or equal to the original number, and the list is infinite.
36 = 1 × 36
36 = 2 × 18
36 = 3 × 12
36 = 4 × 9
36 = 6 × 6 ← pairs meet; stop here
Testing 5: 36 ÷ 5 = 7.2, not a whole number — skip.
List all factors of each number using factor pairs.
Prime number: A whole number greater than 1 with exactly two factors: 1 and itself.
Composite number: A whole number greater than 1 with more than two factors.
Special cases: The numbers 0 and 1 are neither prime nor composite.
The first several prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Each of these numbers can only be divided evenly by 1 and by itself.
The number 2 is unique: it is even, but its only factors are 1 and 2. It is the only even prime number. Every other even number is divisible by 2, giving it at least three factors, which makes it composite.
Testing whether a number is prime requires checking whether any number from 2 up to its square root divides it evenly. If none do, the number is prime.
1 → factors: {1} → Neither
13 → factors: {1, 13} → Prime
28 → factors: {1,2,4,7,14,28} → Composite
41 → factors: {1, 41} → Prime
60 → many factors → Composite
It is worth memorizing the prime numbers up to at least 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. This will speed up your work in factoring, GCF, LCM, and fraction simplification throughout the rest of this textbook.
Classify each number as prime, composite, or neither. If composite, list its factors.
Every composite number can be written as a product of prime numbers. This is called the prime factorization of the number. The primes in the product are called its prime factors. There is a foundational principle in mathematics called the Fundamental Theorem of Arithmetic: every whole number greater than 1 either is a prime, or can be expressed as a product of prime numbers in exactly one way (ignoring the order of the factors). This means every number has a unique prime factorization.
When a prime appears more than once in a factorization, use an exponent to write it compactly. For example: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 is prime — stop.
So: 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
Writing 48 = 4 × 12 is a factorization, but it is not a prime factorization because 4 and 12 are composite. You must continue factoring every composite factor until every factor in the product is prime.
A factor tree is a visual method for finding the prime factorization of a number. It works by repeatedly breaking a number into two factors until every branch ends in a prime number. You can start a factor tree with any pair of factors. The tree will look different depending on your starting pair, but the prime factorization at the end will always be the same — confirming the Fundamental Theorem of Arithmetic.
1. Write the number at the top.
2. Write any two factors that multiply to give that number.
3. Circle any factor that is already prime.
4. Continue factoring any composite factor.
5. When all branches end in primes, collect the circled primes — that is the prime factorization.
60
/ \
6 10
/ \ / \
2 3 2 5
Primes collected: 2, 3, 2, 5
60
/ \
4 15
/ \ / \
2 2 3 5
Primes collected: 2, 2, 3, 5
180
/ \
2 90
/ \
2 45
/ \
9 5
/ \
3 3
Primes collected: 2, 2, 3, 3, 5
So: 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
If you are not sure where to begin, always try dividing by 2 first (if the number is even), then 3 (using the digit-sum rule), then 5, then 7. This gives you a reliable starting path for virtually any number you will encounter in this course.
Write the prime factorization using exponents. Show your factor tree or division steps.
The Greatest Common Factor (GCF) of two or more whole numbers is the largest whole number that divides evenly into all of them. You may also see this called the Greatest Common Divisor (GCD) or Highest Common Factor (HCF). The GCF is the tool you use when you want to reduce or simplify something — dividing into equal parts, writing fractions in lowest terms, or factoring out the shared component of two quantities.
List all factors of each number. Identify the factors that appear in every list. The greatest of those common factors is the GCF.
1. Write the prime factorization of each number.
2. Identify the prime factors common to all numbers.
3. For each common prime, take the smallest exponent that appears across all factorizations.
4. Multiply these selected powers together.
60 = 2² × 3 × 5
90 = 2 × 3² × 5
Common primes: 2, 3, 5
Smallest power of 2: min(2,1) = 1 → 2¹
Smallest power of 3: min(1,2) = 1 → 3¹
Smallest power of 5: min(1,1) = 1 → 5¹
GCF = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30
24 = 2³ × 3
36 = 2² × 3²
48 = 2⁴ × 3
Common primes: 2 and 3
Smallest power of 2: min(3,2,4) = 2 → 2²
Smallest power of 3: min(1,2,1) = 1 → 3¹
GCF = 2² × 3 = 4 × 3 = 12
Two numbers are called relatively prime (or coprime) if their GCF is 1. This means they share no common factors other than 1. For example, GCF(8, 15) = 1, so 8 and 15 are relatively prime.
When finding the GCF, take the smallest exponent for each common prime. When finding the LCM (next section), take the largest exponent. Mixing these up is the most common error students make in this chapter.
The Least Common Multiple (LCM) of two or more whole numbers is the smallest positive whole number that is a multiple of all of them. The LCM is the tool you use when you want to combine or align two things — finding the Least Common Denominator for fractions, determining when two repeating events will coincide, or finding the smallest quantity that fills multiple containers exactly.
1. Write the prime factorization of each number.
2. Identify every prime that appears in any factorization.
3. For each prime, take the largest exponent that appears across all factorizations.
4. Multiply these selected powers together.
18 = 2 × 3²
24 = 2³ × 3
All primes that appear: 2, 3
Largest power of 2: max(1,3) = 3 → 2³
Largest power of 3: max(2,1) = 2 → 3²
LCM = 2³ × 3² = 8 × 9 = 72
4 = 2²
6 = 2 × 3
10 = 2 × 5
All primes: 2, 3, 5
Largest power of 2: 2²
Largest power of 3: 3¹
Largest power of 5: 5¹
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60
For any two numbers a and b: GCF(a, b) × LCM(a, b) = a × b. For example, GCF(4, 6) = 2 and LCM(4, 6) = 12. Check: 2 × 12 = 24 = 4 × 6. This relationship gives you a quick way to verify your answers.
The GCF and LCM look similar in how they are found, and students frequently confuse which one to use. The key is understanding what each tool does. A useful way to remember: GCF goes down (reduces, divides, simplifies), and LCM goes up (extends, combines, aligns).
| Use the GCF when… | Use the LCM when… |
|---|---|
| You want to reduce or simplify something | You want to combine or align two things |
| You want the largest equal-sized groups | You want the smallest shared unit |
| Problems say "greatest," "largest," "most" | Problems say "smallest," "least," "first time" |
| Dividing items into identical packages | Finding when repeating events coincide |
| Simplifying fractions to lowest terms | Finding the Least Common Denominator |
GCF(24, 36) = 12She can make 12 bags. Each bag gets 24 ÷ 12 = 2 chocolate and 36 ÷ 12 = 3 vanilla cookies.LCM(8, 12) = 24Many students automatically compute whichever of GCF or LCM comes to mind first without reading the problem carefully. Slow down. Ask: is this problem about dividing and reducing (GCF), or about extending and aligning (LCM)?
For each problem, state whether GCF or LCM applies, then solve.
A fraction is in lowest terms (also called simplest form) when the numerator and denominator share no common factors other than 1. To reduce a fraction to lowest terms, divide both the numerator and denominator by their GCF. Using the GCF does it in a single step; dividing by smaller common factors works too but takes more steps.
When adding or subtracting fractions with different denominators, you need a common denominator. The Least Common Denominator (LCD) is the LCM of the denominators. Using the LCD produces the simplest common denominator and avoids unnecessarily large numbers.
45 = 3² × 5
60 = 2² × 3 × 5
GCF(45, 60) = 3 × 5 = 15
45/60 = (45 ÷ 15)/(60 ÷ 15) = 3/4
12 = 2² × 3
8 = 2³
LCM(12, 8) = 2³ × 3 = 24 ← LCD
5/12 = 10/24
3/8 = 9/24
10/24 + 9/24 = 19/24
The arithmetic of factors and multiples appears constantly in everyday reasoning: dividing things equally, scheduling, measurement, construction, and cooking. The key is translating the situation into a GCF or LCM question. When you read a word problem, ask yourself whether you are looking for a shared maximum (GCF) or a shared minimum (LCM).
48 = 2⁴ × 3
60 = 2² × 3 × 5
GCF(48, 60) = 2² × 3 = 12
Pieces from 48-inch ribbon: 48 ÷ 12 = 4
Pieces from 60-inch ribbon: 60 ÷ 12 = 5
15 = 3 × 5
20 = 2² × 5
LCM(15, 20) = 2² × 3 × 5 = 60
36 = 2² × 3²
54 = 2 × 3³
GCF(36, 54) = 2 × 3² = 18
Tiles along width: 36 ÷ 18 = 2
Tiles along length: 54 ÷ 18 = 3
Total tiles: 2 × 3 = 6
Use the Read → Represent → Solve → Check → Explain method.
In this chapter, you worked with factors of specific whole numbers. In Algebra I, you will apply the same idea to expressions involving variables. The arithmetic you practiced here does not disappear — it becomes the engine inside algebraic manipulation.
| Arithmetic | Algebraic Analogue |
|---|---|
12 = 4 × 3 | 12x = 4 × 3x — the factor 4 carries forward |
GCF(6, 9) = 3 | 6x + 9 = 3(2x + 3) — factor out the GCF |
| Combining like terms | 3x + 5x = 8x — both terms are multiples of x |
| Prime numbers (unfactorable) | Prime-like expressions that cannot be factored further |
When you factor the algebraic expression 6x + 9, the first step is finding GCF(6, 9) = 3, then writing 3(2x + 3). If you understand GCF well from arithmetic, this algebraic step will feel familiar rather than frightening. The skill you are building now transfers directly.
Similarly, when you combine like terms in algebra — adding 3x + 5x to get 8x — you are using the distributive property in reverse, factoring out the x. Both terms are multiples of x, just as both 15 and 20 are multiples of 5.
Answer each question in one to three complete sentences.
Test each number for divisibility by 2, 3, 5, 9, and 10. State which apply.
If false, correct the statement.
Write the prime factorization using exponents.
Use a factor tree to find the prime factorization. Show your work.
Classify each as prime or composite. If composite, write its prime factorization.
For each problem, use the Read → Represent → Solve → Check → Explain method. State clearly whether you are finding a GCF or LCM.
Each item contains an error. Identify the error and write the correct solution.
Problem: Find the GCF of 36 and 48.
Student's Work:
36 = 2² × 3²
48 = 2⁴ × 3
GCF = 2⁴ × 3² = 16 × 9 = 144
What error did the student make? What is the correct answer?
Problem: Is 51 a prime number?
Student's Work:
"51 ends in 1, which is odd, so it can't be divided
by 2. No other small numbers seem to work. 51 must be prime."
What test did the student neglect? Is 51 prime or composite? Show the correct test.
Problem: Find the LCM of 9 and 12.
LCM = 9 × 12 = 108The product method is not always wrong, but it is inefficient and produces an incorrect LCM here. What is the correct LCM of 9 and 12? When does LCM = a × b, and when does it not?
Problem: Simplify 36/48.
Student's Work:
"Divide top and bottom by 4:
36/48 = 9/12. Done."
The student simplified using a common factor of 4, but the fraction is not yet in lowest terms. What should the student have done? What is the fully simplified fraction?
Student claim: "2 is composite because it can be divided by 1 and 2, and that's two factors."
Correct this reasoning. Why is 2 prime, not composite?
Chapter 2 gave you the vocabulary and tools of number structure. You can now look at any whole number and describe its prime factors, its complete factor list, and its relationship to other numbers through GCF and LCM.
Chapter 3 will use these tools immediately and repeatedly. Every fraction simplification in that chapter depends on GCF. Every common denominator depends on LCM. Chapter 6 will use ratio and proportion, where common factors help simplify rates and comparisons. Chapter 9 will introduce variables, where factoring means pulling out a GCF from an algebraic expression rather than a number.
The ideas from this chapter do not disappear. They go beneath the surface for a few chapters and then return, doing real work. Invest the time to be comfortable with prime factorization, GCF, and LCM now, and the chapters ahead will be considerably more manageable.