Lesson 4: Prime Factorization

Property

When we see a multiplication of whole numbers, such as $a \cdot b = c$ we say that $a$ and $b$ are a factor pair for $c$.
To test if $a$ is a factor of $c$, we divide $c$ by $a$. If the remainder is zero, $a$ is a factor of $c$.
A prime number is a whole number greater than 1 that has only two distinct factors, 1 and itself.
A number which is not prime is said to be composite.

Examples

• The factor pairs of 18 are $1 \times 18$, $2 \times 9$, and $3 \times 6$. The factors are 1, 2, 3, 6, 9, 18. Since it has more than two factors, 18 is composite.
• The number 13 is prime because its only factors are 1 and 13.
• The number 25 is composite because it has factors 1, 5, and 25. It can be written as $5 \times 5$.

Explanation

Factors are numbers you multiply to get another number. Primes are special because they only have two factors: 1 and themselves. All other numbers greater than 1 are composite, built from multiplying other numbers together.

Property

To find all factor pairs of a number $n$, test divisors from $1$ up to $\sqrt{n}$.
For each divisor $d$ that divides $n$ evenly, the factor pair is $(d, \frac{n}{d})$.

Examples

Property

A factor tree systematically breaks down a composite number into prime factors by repeatedly dividing by the smallest possible prime factor until only prime numbers remain.
The prime factorization is written as $n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}$ where each $p_i$ is prime.

Examples