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# 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 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