Introduction

Greatest Common Factor and Least Common Multiple is an important Grade 6 math skill because students are moving from simple answers toward explaining how the math works.

In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with greatest common factor and least common multiple.

What Is Greatest Common Factor and Least Common Multiple?

Greatest Common Factor and Least Common Multiple means choosing a model, naming what each number means, and explaining the strategy.

The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.

Understanding Greatest Common Factor and Least Common Multiple

Before solving, students should slow down and decide what each number, shape, unit, or label represents.

  • Read the question carefully and identify what is being asked.
  • Choose a model, equation, table, or diagram that matches the situation.
  • Solve one step at a time and keep units or labels attached.
  • Use the answer explanation to check that the result makes sense.

Visual Models

Visual Model 1

Question: Find the LCM of \(12\) and \(15\) using the table below.

Multiples of 12122436486072
Multiples of 15153045607590
  • A. \(30\)
  • B. \(36\)
  • C. \(48\)
  • D. \(60\)

Why it works: Looking at the table, the first number that appears in both rows is \(60\). This is the least common multiple.

Answer: \(60\)

Visual Model 2

Question: Using a factor tree, the prime factorization of \(60\) is:

Visual Model 2

  • A. \(2^2 \times 3 \times 5\)
  • B. \(2 \times 3 \times 5\)
  • C. \(2 \times 3 \times 10\)
  • D. \(2^3 \times 3\)

Why it works: Following the tree: \(60 = 6 \times 10 = (2 \times 3) \times (2 \times 5) = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5\).

Answer: \(2^2 \times 3 \times 5\)

Worked Examples

Example 1

Question: What is the greatest common factor (GCF) of \(24\) and \(36\)?

  • A. \(6\)
  • B. \(8\)
  • C. \(12\)
  • D. \(24\)
  1. List what both numbers share evenly: candidates include \(12\), since \(24 = 12 \times 2\) and \(36 = 12 \times 3\).
  2. Nothing larger divides both cleanly, so the GCF settles at \(12\).

Answer: \(12\)

Example 2

Question: Which list contains all factors of \(20\)?

  • A. \(1, 2, 4, 5, 10, 20\)
  • B. \(1, 2, 3, 4, 5, 20\)
  • C. \(2, 4, 5, 10\)
  • D. \(1, 2, 4, 5, 8, 20\)
  1. When we divide \(20\) by each number: \(20 \div 1 = 20\), \(20 \div 2 = 10\), \(20 \div 4 = 5\), \(20 \div 5 = 4\), \(20 \div 10 = 2\), \(20 \div 20 = 1\).
  2. All these are factors.

Answer: \(1, 2, 4, 5, 10, 20\)

Example 3

Question: What is the least common multiple (LCM) of \(6\) and \(8\)?

  • A. \(24\)
  • B. \(18\)
  • C. \(14\)
  • D. \(48\)
  1. Multiples of \(6\): \(6, 12, 18, 24, 30, \ldots\).
  2. Multiples of \(8\): \(8, 16, 24, 32, \ldots\).
  3. The least common multiple is \(24\).

Answer: \(24\)

Real-World Word Problems

Problem 1

Question: A teacher wants to divide \(24\) students into equal groups with no students left over. Which group size is NOT possible?

  • A. \(3\) students per group
  • B. \(8\) students per group
  • C. \(6\) students per group
  • D. \(5\) students per group

Why it works: The factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12, 24\). Since \(5\) is not a factor of \(24\), we cannot divide equally into groups of \(5\).

Answer: \(5\) students per group

Problem 2

Question: Bus Route A stops every \(8\) minutes. Bus Route B stops every \(12\) minutes. If both buses just left the station, when will they both be at the station again at the same time?

  • A. In \(8\) minutes
  • B. In \(12\) minutes
  • C. In \(20\) minutes
  • D. In \(24\) minutes

Why it works: This is an LCM problem. Multiples of \(8\): \(8, 16, 24, \ldots\). Multiples of \(12\): \(12, 24, \ldots\). Both buses are at the station together at \(24\) minutes.

Answer: In \(24\) minutes

Common Mistakes

  • Rushing before identifying what the numbers represent.
  • Choosing an operation that does not match the situation.
  • Dropping labels, units, or context from the answer.
  • Skipping the estimate or reasonableness check.

Strategy Tips

  • Underline the question being asked.
  • Use a model before jumping to computation.
  • Write an equation that matches the story or picture.
  • Explain the final answer in a sentence.

Practice Questions

Question 1

Which pair of numbers has a GCF of \(5\)?

  • A. \(15\) and \(25\)
  • B. \(10\) and \(20\)
  • C. \(12\) and \(20\)
  • D. \(14\) and \(35\)

Question 2

What is the GCF of \(16\) and \(40\)?

  • A. \(4\)
  • B. \(20\)
  • C. \(16\)
  • D. \(8\)

Question 3

Which is a multiple of both \(4\) and \(6\)?

  • A. \(8\)
  • B. \(12\)
  • C. \(18\)
  • D. \(10\)

Question 4

What is the GCF of \(18\), \(27\), and \(45\)?

  • A. \(3\)
  • B. \(9\)
  • C. \(6\)
  • D. \(18\)

Question 5

Find the LCM of \(5\) and \(9\).

  • A. \(45\)
  • B. \(35\)
  • C. \(40\)
  • D. \(15\)

Question 6

What is the prime factorization of \(30\)?

  • A. \(2 \times 3 \times 5\)
  • B. \(2 \times 15\)
  • C. \(3 \times 10\)
  • D. \(2 \times 2 \times 3\)
Full Answer Explanations Click to show all answers and explanations

Question 1

Answer: \(5\)

Factors of \(15\): \(1, 3, 5, 15\). Factors of \(25\): \(1, 5, 25\). Common factors: \(1, 5\). GCF is \(5\). For option B, GCF is \(10\); for C, GCF is \(4\); for D, GCF is \(7\).

Question 2

Answer: \(8\)

Factors of \(16\): \(1, 2, 4, 8, 16\). Factors of \(40\): \(1, 2, 4, 5, 8, 10, 20, 40\). The greatest common factor is \(8\).

Question 3

Answer: \(12\)

\(12 = 4 \times 3\) and \(12 = 6 \times 2\), so \(12\) is divisible by both \(4\) and \(6\). Option A is a multiple of \(4\) but not \(6\); C is a multiple of \(6\) but not \(4\); D is neither.

Question 4

Answer: \(9\)

Factors of \(18\): \(1, 2, 3, 6, 9, 18\). Factors of \(27\): \(1, 3, 9, 27\). Factors of \(45\): \(1, 3, 5, 9, 15, 45\). Common to all three: \(1, 3, 9\). GCF is \(9\).

Question 5

Answer: \(45\)

Multiples of \(5\): \(5, 10, 15, 20, 25, 30, 35, 40, 45, \ldots\). Multiples of \(9\): \(9, 18, 27, 36, 45, \ldots\). The least common multiple is \(45\) because \(5\) and \(9\) share no common factors.

Question 6

Answer: \(2 \times 3 \times 5\)

\(30 = 2 \times 15 = 2 \times 3 \times 5\). The prime factors are \(2, 3, 5\). Option B uses \(15\), which is composite; C uses \(10\), which is composite; D gives prime factorization of \(12\).

Timed practice quizzes

Related Quizzes

After studying Greatest Common Factor and Least Common Multiple, open these two timed quizzes in a new tab for focused extra practice.

Connection to Standards

This lesson supports Grade 6 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.

Summary

Greatest Common Factor and Least Common Multiple becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.

GOLDEN RULE

Understand the model before choosing the operation.