1. Number Sense

United States of America

Grade 6

All whole numbers except for 1 are either prime or composite, and composite numbers can always be written as a product of primes. Finding this product (called a **prime factoring**) can be very useful.

A number is **prime** if it has exactly two factors: 1, and itself. A number is **composite** if it has more than two factors.

One of the best ways to find a prime factoring is by using a **factor tree**. We start with the number we want to investigate, draw a box around it, and draw two lines coming out of it.

This is a completed factor tree for 12, and it tells us that 12 = 2 \times 2 \times 3. Multiplying the circled numbers at the end of each branch together always makes the original number.

Even though the number in the box is different, the numbers at the end of the branches will always be the same for any number - they will just be in a different order.

We can therefore write:

\displaystyle 360 | \displaystyle = | \displaystyle 3 \times 5 \times 2 \times 2 \times 2 \times 3 | |

\displaystyle 360 | \displaystyle = | \displaystyle 2 \times 2 \times 2 \times 3 \times 3 \times 5 | We usually rewrite the factors in ascending order |

\displaystyle 360 | \displaystyle = | \displaystyle 2^{3} \times 3^{2} \times 5 | We can use exponent notation to make the expression shorter |

Notice that the factor tree for 12 we made earlier is a smaller part of the factor tree for 360. This is because 12 is a factor of 360, and when we write 360 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 we can recognize the prime factoring of 12 inside it: 360 = 2 \times (2 \times 2 \times 3) \times 3 \times 5.

A number has the following factor tree:

What is this number at the top of the tree?

Worked Solution

Write 144 as a product of prime factors in expanded form.

Worked Solution

Idea summary

A number is a **prime** if it has exactly two factors: 1, and itself.

A number is **composite** if it has more than two factors.

A **factor tree** starts with the number that needs to be investigated and branches out to two factors. Each composite number in the factor tree branches out to two more factors until the last row of the tree are all primes.