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What is the largest factor of \(27\) less than \(27\)?

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Explanation: Factors: \(1,3,9,27\); largest less than 27 is \(9\).

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Factors & Multiples

Factors and Multiples Practice Quiz with a Step-by-Step Interactive Lesson

Use the quiz at the top of the page to practice factors, multiples, prime and composite numbers, GCF (greatest common factor), and LCM (least common multiple). If you want a refresher, click Start lesson to open a step-by-step guide with examples and quick checks.

How this factors and multiples practice works

1. Take the quiz: answer the questions at the top of the page.
2. Open the lesson (optional): review key methods for listing factors, finding multiples, and solving GCF/LCM problems.
3. Retry: return to the quiz and apply what you reviewed.

What you’ll learn in the factors and multiples lesson

Meaning & vocabulary

Factors (divisors) vs. multiples
Factor pairs and listing factors in order
Prime, composite, and “neither” (the number \(1\))

Listing strategies

How to list all factors using factor pairs
How to generate and count multiples in a range
Common factors and common multiples

GCF and LCM

Greatest common factor (GCF / GCD / HCF)
Least common multiple (LCM)
Using GCF/LCM for simplifying and common denominators

Divisibility & number sense

Divisibility rules for \(2,3,4,5,6,8,9,10\)
Quick checks to decide “factor or not?”
Building strong number sense for mental math

Back to the quiz

When you’re ready, return to the quiz at the top of the page and continue practicing.

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Factors & Multiples
Lesson

Step-by-step guide

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Factors & Multiples Lesson

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Lesson Overview

Lesson overview

Purpose: Build a clear understanding of factors and multiples, and learn reliable methods for GCF, LCM, and prime/composite questions.

Success criteria

Explain what a factor is: a number that divides evenly (no remainder).
List the factors of a number in order using factor pairs.
Generate and count multiples in a range (for example, multiples of 3 up to 18).
Identify common factors and common multiples.
Find GCF (greatest common factor / GCD / HCF) and LCM (least common multiple), including for 3 numbers.
Classify numbers as prime, composite, or neither (the number \(1\)).
Use divisibility rules to test quickly if a number is a factor or multiple.
Use GCF/LCM in real problems: simplifying, common denominators, schedules, and rectangular arrays.

Key vocabulary

Factor (divisor): a number that divides another number evenly.
Multiple: a number that can be written as \(n\times k\) for some whole number \(k\).
Prime number: exactly two positive factors (1 and itself).
Composite number: more than two positive factors.
GCF (GCD/HCF): the greatest common factor of two or more numbers.
LCM: the least common multiple of two or more numbers.

Quick pre-check

Pre-check 1: Which number is a factor of \(12\)?

\(5\) \(6\) \(7\) \(9\)

Hint: A factor divides \(12\) with no remainder. Check \(12\div 6\).

Pre-check 2: How many factors does \(12\) have?

Hint: List them: \(1,2,3,4,6,12\).

Factors

Factors and factor pairs

Learning goal: Find all factors of a number and list them in order using factor pairs.

Key idea

A factor of a number \(n\) is a whole number that divides \(n\) evenly. In other words, \(a\) is a factor of \(n\) if \(n \div a\) has no remainder. Factors come in pairs: if \(a\) is a factor of \(n\), then there is a number \(b\) such that \(a\times b = n\).

Worked example

Example: List the factors of \(24\)

Find factor pairs that multiply to 24:
\(1\times 24\), \(2\times 12\), \(3\times 8\), \(4\times 6\).
So the factors of \(24\) in order are:
\(1,2,3,4,6,8,12,24\).

Try it

Try it 1: What is the third factor of \(12\) when listed in order?

Hint: Factors of \(12\) in order: \(1,2,3,4,6,12\).

Try it 2: What is the fourth factor of \(24\) when listed in order?

\(3\) \(4\) \(6\) \(8\)

Hint: Factors of \(24\) in order are \(1,2,3,4,6,8,12,24\).

Summary

A factor divides a number evenly (no remainder).
Use factor pairs to list all factors in order.

Multiples

Multiples and counting multiples

Learning goal: Generate multiples, count multiples in a range, and recognize non-multiples quickly.

Key idea

A multiple of \(n\) is a number you get by multiplying \(n\) by a whole number: \(n\times 1, n\times 2, n\times 3,\dots\). If \(m\) is a multiple of \(n\), then \(n\) is a factor of \(m\).

Worked example

Example: How many multiples of \(3\) are there up to and including \(18\)?

List the multiples: \(3,6,9,12,15,18\).
There are 6 multiples of \(3\) up to \(18\).

Try it

Try it 1: How many multiples of \(5\) are there up to and including \(20\)?

Hint: Multiples of \(5\) up to \(20\) are \(5,10,15,20\).

Try it 2: Which of these is NOT a multiple of \(3\)?

\(9\) \(12\) \(14\) \(18\)

Hint: Multiples of \(3\) are divisible by \(3\) (no remainder).

Summary

Multiples are made by multiplying by whole numbers: \(n,2n,3n,\dots\).
To count multiples up to a limit, you can list them or use division carefully.

Prime & Composite

Prime numbers, composite numbers, and prime factorization

Learning goal: Classify numbers as prime or composite and use prime factors to describe composites.

Key idea

A prime number has exactly two positive factors: \(1\) and itself. A composite number has more than two positive factors. The number \(1\) is neither prime nor composite.

Worked example

Example: Is \(27\) prime or composite?

Check small factors: \(27\div 3 = 9\), so \(3\) is a factor of \(27\).
That means \(27\) has factors other than \(1\) and \(27\), so it is composite.
Prime factorization: \(27 = 3\times 3\times 3 = 3^3\).

Try it

Try it 1: Which of these is prime: \(8, 9, 10,\) or \(11\)?

\(8\) \(9\) \(10\) \(11\)

Hint: A prime number has exactly two factors: \(1\) and itself.

Try it 2: Which of these is composite?

\(2\) \(3\) \(4\) \(5\)

Hint: A composite number has more than two factors.

Summary

Prime: exactly two factors. Composite: more than two factors.
Prime factorization writes a composite as a product of primes (for example, \(27=3^3\)).

Greatest Common Factor

Common factors and the greatest common factor (GCF)

Learning goal: Find common factors and identify the greatest common factor (also called GCD or HCF).

Key idea

Common factors are factors shared by two numbers. The greatest common factor (GCF) is the largest factor they share. You can find it by listing factors or by using prime factorization.

Worked example

Example: Find the GCF of \(8\) and \(12\)

Factors of \(8\): \(1,2,4,8\).
Factors of \(12\): \(1,2,3,4,6,12\).
Common factors: \(1,2,4\).
So the GCF is \(4\). There are 3 common factors.

Try it

Try it 1: How many common factors do \(8\) and \(12\) have?

Hint: The common factors are \(1,2,4\).

Try it 2: What is the greatest common factor of \(14\) and \(28\)?

Hint: If one number is a multiple of the other, the smaller number is the GCF.

Summary

Common factors are shared factors.
The GCF is the largest common factor.

Least Common Multiple

Common multiples and the least common multiple (LCM)

Learning goal: Find common multiples and identify the least common multiple (LCM).

Key idea

Common multiples are multiples shared by two or more numbers. The least common multiple (LCM) is the smallest positive number that is a multiple of each number. You can find it by listing multiples or using prime factorization.

Worked example

Example: Find the LCM of \(2\) and \(3\)

Multiples of \(2\): \(2,4,6,8,\dots\)
Multiples of \(3\): \(3,6,9,12,\dots\)
The smallest common multiple is \(6\), so \(\mathrm{LCM}(2,3)=6\).

Try it

Try it 1: What is the smallest common multiple of \(2\) and \(3\)?

Hint: List multiples until you see a match: \(2,4,6,\dots\) and \(3,6,\dots\).

Try it 2: What is the least common multiple of \(2, 3,\) and \(5\)?

Hint: \(30\) is divisible by \(2\), \(3\), and \(5\).

Summary

Common multiples are shared multiples.
The LCM is the smallest positive common multiple.

Divisibility Rules

Divisibility rules to test factors and multiples quickly

Learning goal: Use divisibility rules as fast checks for factor and multiple questions.

Key idea

Divisibility rules help you decide quickly if a number divides evenly. Here are some common rules:

Divisible by 2: last digit is even (0,2,4,6,8).
Divisible by 3: sum of digits is divisible by 3.
Divisible by 4: last two digits form a number divisible by 4.
Divisible by 5: last digit is 0 or 5.
Divisible by 6: divisible by 2 and by 3.
Divisible by 9: sum of digits is divisible by 9.
Divisible by 10: last digit is 0.

Worked example

Example: Is \(4\) a factor of both \(12\) and \(20\)?

Check \(12\div 4 = 3\) (no remainder) and \(20\div 4 = 5\) (no remainder).
So yes: \(4\) is a factor of both \(12\) and \(20\).

Try it

Try it 1: Is \(4\) a factor of both \(12\) and \(20\)?

Yes No

Hint: Divide \(12\) and \(20\) by \(4\). If there is no remainder, then \(4\) is a factor.

Try it 2: Which of these is NOT a multiple of \(6\)?

\(12\) \(18\) \(20\) \(24\)

Hint: A multiple of \(6\) is divisible by \(6\) with no remainder.

Summary

Divisibility rules help you test quickly without long division.
Use them to decide if a number is a factor or a multiple.

Applications & Review

Why factors and multiples matter

Learning goal: Connect factors and multiples to simplifying, schedules, and everyday math — then review key skills.

Where you use factors and multiples

Simplifying: use the GCF to simplify fractions and ratios.
Common denominators: use the LCM to add and subtract fractions with different denominators.
Schedules: repeating events meet again after an LCM amount of time.
Arrays and rectangles: factor pairs describe possible row-and-column arrangements.

Worked example: repeating events (LCM)

Example: One bell rings every 6 minutes and another rings every 8 minutes. When will they ring together?

This is a least common multiple problem:
\(\mathrm{LCM}(6,8)=24\).
Answer: They ring together every \(24\) minutes.

Try it

Try it 1: How many multiples of \(7\) are there up to and including \(28\)?

Hint: Multiples of \(7\) up to \(28\) are \(7,14,21,28\).

Try it 2: Which number is both a factor of \(12\) and a multiple of \(3\)?

\(4\) \(6\) \(9\) \(10\)

Hint: Factors of \(12\) are \(1,2,3,4,6,12\). Multiples of \(3\) are \(3,6,9,12,\dots\).

Final recap

Factors divide evenly; multiples come from multiplying by whole numbers.
Use factor pairs to list factors and keep them in order.
Prime vs. composite: prime has exactly two factors; composite has more than two.
GCF is the biggest shared factor; LCM is the smallest shared multiple.
Divisibility rules help you check quickly and reduce mistakes.

Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the skill (factors, multiples, prime/composite, GCF, or LCM).