## Learning Math: Number and Operations

# Number Theory

## Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.

**In This Session:**

Part A:Models for Multiples and Factors

Part A:

**Part B:**Looking for Prime Numbers

**Homework**

As part of our exploration of number theory, we will look at two models for finding least common multiples and greatest common factors: the Venn diagram model and the area model. Later in the session, we will explore prime and composite numbers.

For information on required and/or optional material for this session, see Note 1.

**LEARNING OBJECTIVES
**In this session, you will do the following:

• Understand greatest common factors and least common multiples and how they relate to one another

• Understand alternative models and methods for computing greatest common factors and least common multiples

• Understand prime and composite numbers

• Understand the location of prime numbers within the number system, and use this understanding to determine whether very large numbers are prime

### Key Terms

**Previously Introduced**

**counting numbers**

Counting numbers are the same as natural numbers (i.e., 1, 2, 3, 4, …).

**factor**

A factor of a number is a counting number that divides evenly into that number. For example, 3 is a factor of 15, since 3 divides evenly into 15 (five times). Four is not a factor of 15, but it is a factor of 16.

**factor tree**

A factor tree can be used to factor a number into prime factors. To create a factor tree, start with the smallest prime factor of the given number and then split the number into factors. With 30, the smallest prime factor is 2, so 30 = 2 • 15. Then factor 15 into prime numbers: 30 = 2 • 15 and 15 = 3 • 5. So, 30 = 2 • 3 • 5, which is its prime factorization.

**prime number**

A counting number is a prime number if it has exactly two factors: 1 and the number itself. For example, 17 is prime, 16 is not prime, and 1 itself is not prime, since it has only one factor.

**New in This Session**

composite number

A counting number is called a composite number if it has more than two factors. For example, 16 is composite because it has five factors (1, 2, 4, 8, and 16).

**greatest common factor**

The greatest common factor of two numbers is the largest number that is a factor of both given numbers. For example, 4 is the greatest common factor of 20 and 28, since it is a factor of both 20 and 28, and no number larger than 4 is a factor of both.

**least common multiple**

The least common multiple of two numbers is the smallest number that is a multiple of both given numbers. For example, 56 is the least common multiple of 8 and 14, since 8 and 14 are each factors of 56, and no number smaller than 56 has both 8 and 14 as factors.

**Venn diagram**

A Venn diagram is a graphic representation of sets. It can be used to show the union and intersection of two sets.

### Notes

**Note 1**

**Optional Material
• **Graph paper for the non-interactive activity