Greatest Common Factor Least Common Multiple Worksheet

Greatest Common Factor (GCF) and Least Common Multiple (LCM) Worksheet: A Comprehensive Guide

Finding the greatest common factor (GCF) and least common multiple (LCM) might seem like a dry math topic, but mastering these concepts is crucial for simplifying fractions, solving algebraic equations, and even tackling more advanced mathematical problems. This comprehensive guide will not only explain GCF and LCM but also provide you with a wealth of practice problems, making your worksheet experience both effective and engaging.

Understanding Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that divides exactly into two or more numbers without leaving a remainder. Imagine you have a collection of apples and oranges, and you want to divide them into equal groups with the same number of apples and oranges in each group. The GCF will tell you the largest possible size of those groups.

Methods for Finding GCF:

There are several ways to find the GCF:

1. Listing Factors:

This method involves listing all the factors of each number and then identifying the largest factor they have in common.

Example: Find the GCF of 12 and 18.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18

The common factors are 1, 2, 3, and 6. The GCF is 6.

2. Prime Factorization:

This method involves breaking down each number into its prime factors (numbers divisible only by 1 and themselves). Then, you multiply the common prime factors to find the GCF.

Example: Find the GCF of 24 and 36.

• Prime factorization of 24: 2 x 2 x 2 x 3 = 2³ x 3
• Prime factorization of 36: 2 x 2 x 3 x 3 = 2² x 3²

The common prime factors are 2² and 3. Therefore, the GCF is 2 x 2 x 3 = 12.

3. Euclidean Algorithm:

This is a more efficient method for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCF.

Example: Find the GCF of 48 and 18.

1. Divide 48 by 18: 48 = 2 x 18 + 12
2. Divide 18 by the remainder 12: 18 = 1 x 12 + 6
3. Divide 12 by the remainder 6: 12 = 2 x 6 + 0

The last non-zero remainder is 6, so the GCF is 6.

Understanding Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. Think of it as finding the smallest number that both of your numbers can divide into evenly. If you are planning events that repeat at different intervals, the LCM will tell you when they will coincide again.

Methods for Finding LCM:

Like GCF, there are multiple ways to find the LCM:

1. Listing Multiples:

This method involves listing the multiples of each number until you find the smallest multiple they have in common.

Example: Find the LCM of 4 and 6.

• Multiples of 4: 4, 8, 12, 16, 20...
• Multiples of 6: 6, 12, 18, 24...

The smallest common multiple is 12, so the LCM is 12.

2. Prime Factorization:

This method is often more efficient, especially for larger numbers. You find the prime factorization of each number, and then you take the highest power of each prime factor present in either factorization.

Example: Find the LCM of 12 and 18.

• Prime factorization of 12: 2² x 3
• Prime factorization of 18: 2 x 3²

The highest power of 2 is 2², and the highest power of 3 is 3². Therefore, the LCM is 2² x 3² = 36.

3. Using the GCF:

There's a handy relationship between GCF and LCM:

LCM(a, b) = (a x b) / GCF(a, b)

This formula allows you to calculate the LCM quickly once you've found the GCF.

Example: Find the LCM of 12 and 18. We already know the GCF(12, 18) = 6.

LCM(12, 18) = (12 x 18) / 6 = 36

Practice Problems: GCF and LCM Worksheet

Now, let's put our knowledge into practice with some examples. Try to solve these problems using the methods described above.

Section 1: GCF

1. Find the GCF of 15 and 25.
2. Find the GCF of 24 and 36.
3. Find the GCF of 48, 72, and 96.
4. Find the GCF of 105 and 168.
5. Find the GCF of 126, 198, and 231.

Section 2: LCM

1. Find the LCM of 8 and 12.
2. Find the LCM of 15 and 20.
3. Find the LCM of 24 and 36.
4. Find the LCM of 35 and 49.
5. Find the LCM of 48, 72, and 96.

Section 3: Mixed Problems

1. Find the GCF and LCM of 18 and 24.
2. Two runners are running on a track. One runner completes a lap every 4 minutes, and the other runner completes a lap every 6 minutes. After how many minutes will they both be at the starting line at the same time? (Hint: Use LCM).
3. A rectangular garden is 24 feet long and 36 feet wide. What is the largest square tile that can be used to completely cover the garden without any cutting? (Hint: Use GCF).
4. Find the GCF and LCM of 108 and 144.
5. A florist has 24 red roses, 36 white roses, and 48 pink roses. She wants to make bouquets with the same number of each color of rose in each bouquet. What is the largest number of bouquets she can make? (Hint: Use GCF).

Answers and Explanations

Section 1: GCF

1. GCF(15, 25) = 5 (Factors of 15: 1, 3, 5, 15; Factors of 25: 1, 5, 25)
2. GCF(24, 36) = 12 (Prime factorization: 24 = 2³ x 3; 36 = 2² x 3²)
3. GCF(48, 72, 96) = 24 (Prime factorization: 48 = 2⁴ x 3; 72 = 2³ x 3²; 96 = 2⁵ x 3)
4. GCF(105, 168) = 21 (Prime factorization: 105 = 3 x 5 x 7; 168 = 2³ x 3 x 7)
5. GCF(126, 198, 231) = 21 (Prime factorization: 126 = 2 x 3² x 7; 198 = 2 x 3² x 11; 231 = 3 x 7 x 11)

Section 2: LCM

1. LCM(8, 12) = 24 (Multiples of 8: 8, 16, 24; Multiples of 12: 12, 24)
2. LCM(15, 20) = 60 (Prime factorization: 15 = 3 x 5; 20 = 2² x 5)
3. LCM(24, 36) = 72 (Prime factorization: 24 = 2³ x 3; 36 = 2² x 3²)
4. LCM(35, 49) = 245 (Prime factorization: 35 = 5 x 7; 49 = 7²)
5. LCM(48, 72, 96) = 288 (Prime factorization: 48 = 2⁴ x 3; 72 = 2³ x 3²; 96 = 2⁵ x 3)

Section 3: Mixed Problems

1. GCF(18, 24) = 6; LCM(18, 24) = 72
2. LCM(4, 6) = 12 minutes
3. GCF(24, 36) = 12 feet The largest square tile would be 12 feet by 12 feet.
4. GCF(108, 144) = 36; LCM(108, 144) = 432
5. GCF(24, 36, 48) = 12 bouquets

This extensive worksheet and solution guide provides a solid foundation in understanding and applying GCF and LCM. Remember to practice regularly to build your skills and confidence in tackling more complex mathematical problems. Consistent practice is key to mastering these essential concepts. Remember to use different methods to find the GCF and LCM to reinforce your understanding and identify the most efficient approach for each problem. Good luck!