Prime Factors -

Factoring whole numbers

To factor an whole number means to write it as the mutliplication of two or more factors.

Let’s factor the integer 24.

$24=12 \times 2$

or,

$24= 8 \times 3$

Here’s another way to factor 24, this time with three factors:

$24=2 \times 2 \times 6$

What is the longest line of factors we can write for 24, not including the number 1?

$24 = 2 \times 2 \times 2 \times 3$

This is the longest line. We know its longest because each of the factors in the line are prime numbers – they can’t be broken down into smaller factor pairs.

A prime number is a whole number that has exactly two factors: 1 and itself.

The first ten prime numbers are:

$2, 3, 5, 7, 11, 13, 17, 19, 23,29\dots$

A whole number bigger than 1 that is not prime is called composite.

We say, the prime factorization of 24 is $2 \times 2 \times 2 \times 3$ or $2^3 \times 3$ .

Let’s factor the integer 40.

We know that $40 = 4 \times 10$ . We also know that $4=2\times2$ and that $10 =2 \times 5$ .

So we have $40 = 4 \times 10 = (2 \times 2) \times (2 \times 5) = 2^3 \times 5$ .

Prime Factor Trees

Prime numbers are a building block to online encription. Read about the RSA cryptosystem here. Any secure websites (emails, bank transactions) use encryption based on prime numbers.

Prime factor trees help to find the prime factorization of any integer. Here is an example of a prime factor tree:

Click on mathisfun to see this described again, or on Khan Academy for a video.

Practice

Practice drawing prime factor trees. Write a conclusion under each tree, for example, $48 = 2 \times 2 \times 2 \times 2 \times 3$ .

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