Number theory—the study of integer numbers—has fascinated mathematicians for years. Fundamental to number theory are whole numbers: 0, 1, 2, 3, and so on. The basic building blocks for all whole numbers are prime numbers. A prime number is a whole number that has only two factors: itself and one. When a number has more than two factors, it is said to be a composite number.
The numbers one and zero are unusual cases. The number one has only one factor and is considered to be neither prime nor composite. The number zero has an infinite number of factors, as zero can be divided evenly by anything except zero. For this reason, zero is also considered neither prime nor composite.
Check Your Understanding:
- Is 36 a prime number or composite number? (Composite, because it has many factors other than 1 and 36: 2, 3, 4, 6, 9, 12, and 18.)
- What are all of the prime numbers less than 10? (2, 3, 5, and 7.)
Prime Factorization of 36
Students in Grades 4 and up are typically ready to learn about prime factors, and students in Grades 6 and up are typically ready to investigate the prime factorization of a number and explore how it can be used to generate all whole numbers.
- Find factor pairs and recognize that a whole number is a multiple of each of its factors.
- Write and evaluate numerical expressions that represent prime factorizations and involve whole-number exponents.
When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. Thus, if we write 36 as a product of all of its prime factors, we can find the prime factorization of 36.
We can write 36 as a product of prime factors: 36 = 2² × 3². The expression 2² × 3² is said to be the prime factorization of 36.
The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every whole number can be factored uniquely (except for the order of the factors) into a product of prime factors. For example, there are many whole numbers that can divide 36: 2, 3, 4, 6, 9, 12, and 18. But no matter which factor you start with, when the number 36 is factored completely, it will always look like 2² × 3². The order in which you factor 36 doesn't matter. Examine the two factor trees of 36 given below.
Be the first to read the latest from Shaped.