English:Factors, Multiples, and Prime Numbers
Factors, Multiples, and Prime Numbers
Introduction
Numbers have patterns. When you learn to see those patterns, many calculations become easier. In this aiMOOC, you will explore factors, multiples, prime numbers, composite numbers, prime factorization, the greatest common factor, and the least common multiple.
A factor divides a whole number exactly, with no remainder. A multiple is found by multiplying a number by a whole number. A prime number has exactly two positive factors: 1 and itself. These ideas are connected. They help you solve problems about equal groups, repeating events, fractions, patterns, and divisibility.
By the end of this course, you should be able to explain the difference between factors and multiples, identify prime and composite numbers, break a composite number into prime factors, and choose useful strategies for common-factor and common-multiple problems.
Learning Goals
After working through the course, you can:
- Factors and multiples: Find factors, factor pairs, and multiples of whole numbers.
- Divisibility rules: Use simple divisibility tests for 2, 3, 5, 9, and 10.
- Prime numbers: Tell whether a number is prime or composite and explain why.
- Prime factorization: Write a composite number as a product of prime numbers.
- Greatest common factor: Find the greatest factor shared by two numbers.
- Least common multiple: Find the smallest positive multiple shared by two numbers.
- Problem solving: Decide whether a real-life problem is asking about factors, multiples, or primes.
Factors
A factor of a whole number is a whole number that divides it exactly. For example, 3 is a factor of 18 because 18 ÷ 3 = 6. The number 5 is not a factor of 18 because 18 ÷ 5 does not give a whole-number answer.
Factors come in factor pairs. To find the factors of 24, look for multiplication pairs that make 24:
- Multiplication: 1 × 24 = 24
- Multiplication: 2 × 12 = 24
- Multiplication: 3 × 8 = 24
- Multiplication: 4 × 6 = 24
So the positive factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
A useful habit is to search for factor pairs in order. Start with 1, then test 2, 3, 4, and so on. When the pairs begin to repeat in reverse order, you have found them all.
Factors as Arrays and Equal Groups
Imagine 24 counters. You can arrange them as 1 row of 24, 2 rows of 12, 3 rows of 8, or 4 rows of 6. Each rectangular arrangement shows one factor pair of 24.
This is why factor questions often appear in problems about equal groups. If 24 students must stand in equal rows with no one left over, the possible numbers of students per row are factors of 24.
A factor is always less than or equal to the number it divides when we work with positive whole numbers. Every positive whole number has 1 and itself as factors.
Divisibility Shortcuts
A divisibility rule can help you decide whether a number has a certain factor without doing long division.
- Divisible by 2: The last digit is 0, 2, 4, 6, or 8.
- Divisible by 3: The sum of the digits is divisible by 3.
- Divisible by 5: The last digit is 0 or 5.
- Divisible by 9: The sum of the digits is divisible by 9.
- Divisible by 10: The last digit is 0.
For example, 126 is divisible by 2 because it is even. It is also divisible by 3 because 1 + 2 + 6 = 9, and 9 is divisible by 3. It is not divisible by 5 because its last digit is neither 0 nor 5.
These rules are tools, not guesses. You can always check by division or multiplication.
Multiples
A multiple of a number is the result of multiplying that number by a whole number. The first few positive multiples of 6 are 6, 12, 18, 24, 30, 36, and so on.
Multiples do not stop. You can always make a larger multiple by multiplying by a larger whole number.
A multiplication chart is also a map of multiples. In the row for 6, you can see multiples of 6. In the row for 8, you can see multiples of 8. A number that appears in both rows is a common multiple.
Factor or Multiple?
The words sound related, but they point in different directions.
Factors build a number. For 18, the factors include 1, 2, 3, 6, 9, and 18.
Multiples grow from a number. Multiples of 18 include 18, 36, 54, 72, and so on.
A quick question can help you choose:
- Factor question: "What whole numbers divide this number exactly?"
- Multiple question: "What numbers do I get when I keep multiplying this number?"
Notice that 6 is a factor of 18, while 18 is a multiple of 6. The relationship works both ways.
Common Multiples
A common multiple is a number that is a multiple of two or more numbers.
For example, multiples of 4 include 4, 8, 12, 16, 20, 24, 28, and 32. Multiples of 6 include 6, 12, 18, 24, 30, and 36. The numbers 12 and 24 are common multiples of 4 and 6.
The smallest positive common multiple is called the least common multiple, often shortened to LCM. For 4 and 6, the LCM is 12.
Prime and Composite Numbers
A prime number is a whole number greater than 1 that has exactly two positive factors: 1 and itself.
Examples include 2, 3, 5, 7, 11, and 13.
A composite number is a whole number greater than 1 that has more than two positive factors. For example, 12 is composite because its factors are 1, 2, 3, 4, 6, and 12.
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The number 2 is special: it is the only even prime number. Every other even number greater than 2 is divisible by 2, so it has at least three factors and is composite.
The number 1 is neither prime nor composite. It has only one positive factor, so it does not meet the definition of a prime number or a composite number.
How to Test Whether a Number Is Prime
For small numbers, you can test possible factors. Suppose you want to decide whether 37 is prime. It is not even, so it is not divisible by 2. The digit sum is 3 + 7 = 10, so it is not divisible by 3. It does not end in 0 or 5, so it is not divisible by 5. No smaller prime factor works, so 37 is prime.
For Grades 5–6, a careful factor search and the divisibility rules are usually enough for numbers of the size you meet in class.
The Sieve of Eratosthenes
The Sieve of Eratosthenes is a systematic way to find prime numbers in a list.
Start with the whole numbers from 2 upward. Circle 2 and cross out its larger multiples. Then circle the next number not crossed out, which is 3, and cross out its larger multiples. Continue with 5, then 7, and so on. The numbers that remain are prime.
Try making your own sieve from 2 to 100. Use a different pencil mark for multiples of 2, 3, 5, and 7. Look for patterns in where the primes appear.
Prime Factorization
Every composite whole number greater than 1 can be written as a product of prime numbers. This is called prime factorization.
For 24, you can split the number into factor pairs and continue splitting every composite factor:
24 = 4 × 6
4 = 2 × 2
6 = 2 × 3
So 24 = 2 × 2 × 2 × 3.
A factor tree is a visual method for doing this. You may start with different factor pairs, but if you continue until every branch ends in a prime number, you reach the same prime factors, apart from their order.
Why Prime Factors Matter
Prime factors are like the basic building blocks of whole numbers. They can help you compare numbers, find common factors, find common multiples, and simplify some fraction problems.
For example:
36 = 2 × 2 × 3 × 3
60 = 2 × 2 × 3 × 5
The shared prime factors are 2 × 2 × 3 = 12. This tells us that 12 is the greatest common factor of 36 and 60.
The image above is a richer visualization of how prime factors appear in many numbers. You do not need to memorize the whole picture. Use it to notice that composite numbers are made from repeated combinations of primes.
Greatest Common Factor
A common factor divides two or more numbers exactly. The greatest common factor, or GCF, is the largest factor the numbers share. It is also called the greatest common divisor.
Suppose you want the GCF of 18 and 24.
Factors of 18 are 1, 2, 3, 6, 9, and 18.
Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The common factors are 1, 2, 3, and 6. The greatest is 6, so GCF of 18 and 24 = 6.
You can find a GCF by listing factors or by comparing prime factorizations.
When a GCF Is Useful
GCF problems often involve making the greatest possible number of equal groups or cutting things into the largest equal pieces with nothing left over.
Example: You have 18 red beads and 24 blue beads. You want to make identical sets using every bead. The greatest number of identical sets is 6, because 6 is the GCF of 18 and 24. Each set gets 3 red beads and 4 blue beads.
Always check what the question asks. A GCF is not automatically the answer just because two numbers appear in a problem.
Least Common Multiple
The least common multiple, or LCM, is the smallest positive number that is a multiple of two or more numbers.
To find the LCM of 8 and 12 by listing:
Multiples of 8 are 8, 16, 24, 32, 40, 48, and so on.
Multiples of 12 are 12, 24, 36, 48, and so on.
The first common multiple is 24, so LCM of 8 and 12 = 24.
You can also use prime factorization:
8 = 2 × 2 × 2
12 = 2 × 2 × 3
To build the smallest number containing all needed prime factors, use three 2s and one 3:
2 × 2 × 2 × 3 = 24.
When an LCM Is Useful
LCM problems often involve repeating events.
Imagine one light flashes every 4 seconds and another flashes every 6 seconds. If they flash together now, they will next flash together after 12 seconds because 12 is the LCM of 4 and 6.
This idea also works for repeating schedules, rotating patterns, exercise routines, and cycles in games.
Choosing the Right Idea
When you meet a number problem, do not rush to a calculation. First ask what the situation means.
- Factors: Use factors when you are dividing one amount into equal whole-number groups or checking exact divisibility.
- GCF: Use a greatest common factor when several amounts must be split into the largest number of equal groups or largest equal pieces.
- Multiples: Use multiples when a quantity grows in repeated equal steps.
- LCM: Use a least common multiple when repeated events must meet again.
- Prime numbers: Use prime ideas when you need to decide whether a number has only two factors or when you want the basic factor building blocks.
A good problem solver explains not only the answer but also why the chosen idea fits the situation.
Worked Examples
Example: Packing Stickers
You have 30 star stickers and want to place the same number on every card with no stickers left over. How many stickers could go on each card?
You need factors of 30. They are 1, 2, 3, 5, 6, 10, 15, and 30. Any of these amounts can make equal groups, depending on how many cards you want.
Example: Two Repeating Bells
One bell rings every 5 minutes. Another rings every 8 minutes. They ring together at 10:00. When will they ring together again?
You need the LCM of 5 and 8. Since 5 and 8 share no prime factors, their LCM is 5 × 8 = 40. They will ring together again 40 minutes later, at 10:40.
Example: Prime or Composite?
Is 51 prime?
The digit sum is 5 + 1 = 6. Because 6 is divisible by 3, 51 is divisible by 3. In fact, 51 = 3 × 17. Therefore, 51 is composite.
Example: Equal Craft Kits
A class has 20 pencils and 30 erasers. The teacher wants to make the greatest possible number of identical kits and use every item.
The problem asks for the GCF of 20 and 30. Their greatest common factor is 10, so the teacher can make 10 kits. Each kit contains 2 pencils and 3 erasers.
Interactive Tasks
Quiz: Test Your Knowledge
Which number is a factor of 28? (7) (!5) (!9) (!12)
Which list shows the first four positive multiples of 6? (6, 12, 18, 24) (!1, 2, 3, 6) (!6, 7, 8, 9) (!12, 18, 24, 30)
Which number is prime? (29) (!21) (!27) (!35)
Which number is composite? (15) (!2) (!7) (!13)
What is special about the number 1? (It is neither prime nor composite) (!It is the smallest prime number) (!It is the smallest composite number) (!It has exactly two positive factors)
What is the prime factorization of 18? (2 × 3 × 3) (!1 × 18) (!2 × 9) (!3 × 6)
What is the greatest common factor of 12 and 18? (6) (!2) (!12) (!36)
What is the least common multiple of 4 and 6? (12) (!2) (!6) (!24)
Which divisibility clue proves that 135 is divisible by 5? (Its last digit is 5) (!Its first digit is 1) (!Its digits add to 8) (!It has three digits)
Two lights flash every 3 seconds and every 5 seconds. After how many seconds will they flash together again? (15 seconds) (!8 seconds) (!10 seconds) (!30 seconds)
Memory Game
| Factor | A whole number that divides another whole number exactly |
| Multiple | A result of multiplying a number by a whole number |
| Prime | A whole number greater than one with exactly two positive factors |
| Composite | A whole number greater than one with more than two positive factors |
| Factorization | Writing a number as a product of factors |
| Divisibility | The property of dividing exactly with no remainder |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Factor | Divides a whole number exactly |
| Multiple | Result of repeated whole-number multiplication |
| Prime number | Has exactly two positive factors |
| Composite number | Has more than two positive factors |
| Special case | Neither prime nor composite |
Match each mathematical term with the correct explanation. Then explain one match to a partner using a multiplication or division sentence.
Crossword Puzzle
| Factor | What do you call a whole number that divides another exactly? |
| Multiple | What do you call a result in a multiplication pattern? |
| Prime | What kind of number has exactly two positive factors? |
| Composite | What kind of number has more than two positive factors? |
| Divisible | What word means able to be divided with no remainder? |
| Factorization | What process writes a number as a product of factors? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Factor hunt: Choose four numbers from 12 to 30, list all factor pairs for each, and check every pair with multiplication.
- Multiple pattern poster: Create a colorful poster showing the first ten positive multiples of 3, 4, and 6 and circle numbers that appear in more than one list.
- Prime number card sort: Make cards for the numbers 2 to 30, sort them into prime and composite groups, and write one reason on the back of each card.
- Divisibility detective: Find ten numbers around you in books, signs, scores, or page numbers and test each for divisibility by 2, 3, 5, and 10.
Standard
- Factor tree gallery: Create factor trees for 24, 36, 48, and 60, then compare the prime factors and write two patterns you notice.
- Sieve of Eratosthenes project: Build a number grid from 2 to 100, carry out the sieve step by step, and design a legend that shows which multiples you removed.
- Common factor interview: Interview a classmate about how they would split two collections into identical groups, record their method, and compare it with your own GCF strategy.
- Repeating events video: Film a short explanation using two repeating actions, such as claps and taps, to show how an LCM predicts when the actions meet again.
Advanced
- School event planner: Design a schedule in which three activities repeat at different intervals, calculate when all three happen together, and justify the result with multiples or prime factors.
- Packaging investigation: Invent a packaging problem with two supplies that must be divided into the greatest possible number of identical kits, solve it with a GCF, and explain why an LCM would not fit.
- Prime pattern investigation: Examine the prime numbers below 100, create a chart or digital image of a pattern you notice, and explain which observations are patterns rather than rules that always work.
- Strategy comparison report: Solve one GCF problem and one LCM problem in two different ways, compare efficiency and clarity, and recommend which method you would teach to another Grade 5–6 learner.
Learning Assessment
- Explain the relationship: Use the numbers 6 and 24 to explain in complete sentences how one number can be a factor of another while the other is a multiple of the first.
- Choose a strategy: A club has 24 juice boxes and 36 snack packs and wants identical sets with nothing left over; decide which number concept is needed, solve the problem, and justify your choice.
- Analyze an error: A learner says that 1 is prime because it can be divided by 1 and itself; explain precisely why this reasoning does not match the definition of a prime number.
- Transfer to schedules: Two buses leave a stop every 12 minutes and every 18 minutes; determine when they next leave together and show at least one method.
- Compare factorizations: Prime-factorize 40 and 60, use the results to find their GCF and LCM, and explain how the shared and unshared prime factors affect each answer.
- Create and defend: Write your own real-life problem whose solution uses either a GCF or an LCM, solve it, and explain why the other operation would not answer the question.
Evidence of Learning
Your work should show both mathematical knowledge and mathematical communication. Strong evidence of learning may include:
- Mathematical knowledge: Accurate factor lists, multiple lists, prime and composite classifications, prime factorizations, GCFs, and LCMs.
- Reasoning: Clear explanations of why a strategy fits a problem and why an answer is reasonable.
- Representation: Useful arrays, factor trees, number grids, tables, or diagrams that make number relationships visible.
- Communication: Correct vocabulary and complete explanations that another learner can follow.
- Problem solving: Successful transfer of factors and multiples to equal-group, packaging, fraction, and repeating-event situations.
- Products: A poster, sieve grid, factor-tree gallery, short video, interview record, investigation chart, or written report from the open-ended tasks.
- Reflection: A short note identifying one strategy that became easier and one idea that still needs practice.
OERs on the Topic
For more reading about prime numbers, you can explore this English Wikipedia article:
You can also connect your learning with divisors, multiples, prime factors, greatest common factors, least common multiples, and arithmetic.
Linked Learning Areas
Factors, multiples, and prime numbers connect multiplication and division with later work on fractions, ratios, algebra, and number theory. Factors help you split quantities exactly. Multiples help you describe repeated steps. Prime numbers provide the basic building blocks for prime factorization. GCF and LCM ideas help you solve equal-group and repeating-cycle problems.