1.06 Prime factorizations

All whole numbers except for $1$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.

Factor trees

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. Here is how we might start with the number $12$12:

We then put two numbers that multiply to make $12$12, such as $4$4 and $3$3, at the end of each of the lines:

Because $3$3 is a prime number, we circle it. Since $4$4 is not a prime number, we draw a box around it instead:

We then repeat the process with $4$4, which is $2\times2$2×2:

And since $2$2 is prime, we circle both of these numbers:

This is a completed factor tree for $12$12, and it tells us that $12=2\times2\times3$12=2×2×3. Multiplying the circled numbers at the end of each branch together always makes the original number.

Factor trees are not always unique - here is another factor tree for $12$12:

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.

Here is a factor tree for $360$360:

We can therefore write: 

$360=3\times5\times2\times2\times2\times3$360=3×5×2×2×2×3

We usually rewrite the factors so they are in ascending order, like this:

$360=2\times2\times2\times3\times3\times5$360=2×2×2×3×3×5

We can use exponent notation to make the expression shorter as well:

$360=2^3\times3^2\times5$360=23×32×5

Notice that the factor tree for $12$12 we made earlier is a smaller part of the factor tree for $360$360. This is because $12$12 is a factor of $360$360, and when we write $360=2\times2\times2\times3\times3\times5$360=2×2×2×3×3×5 we can recognize the prime factoring of $12$12 inside it: $360=2\times\left(2\times2\times3\right)\times3\times5$360=2×(2×2×3)×3×5.

 

Finding all the factors

Factor trees are useful because every number we write as we make it is a factor of the original number. We don't always see every factor appear, though - for example, $9$9 is a factor of $360$360, but it does not appear in the tree above.

To find every factor of a number we need to combine the prime factors in every possible way. First, we find the prime factoring like we did before, such as:

$12=2\times2\times3$12=2×2×3

We then combine these factors in every possible way. Every factor of $12$12 can have no $2$2s, one $2$2, or two $2$2s in its prime factoring. Similarly, every factor of $12$12 can have no $3$3s, or one $3$3. Here we draw this out in a table:

Factors of $12$12
	No $2$2	One $2$2	Two $2$2s
No $3$3	$1$1	$2$2	$2\times2$2×2
One $3$3	$3$3	$2\times3$2×3	$2\times2\times3$2×2×3

We then perform each of the multiplications to find all the factors:

Factors of $12$12
	No $2$2	One $2$2	Two $2$2s
No $3$3	$1$1	$2$2	$4$4
One $3$3	$3$3	$6$6	$12$12

The factors of $12$12 are $1$1, $2$2, $3$3, $4$4, $6$6, and $12$12.

Here is how we can do it for $360$360 - every factor either has $5$5 as a factor or it doesn't, it has between zero and two $3$3s, and between zero and three $2$2s.

Factors of $360$360
		No $2$2	One $2$2	Two $2$2s	Three $2$2s
No $5$5	No $3$3	$1$1	$2$2	$2\times2$2×2	$2\times2\times2$2×2×2
	One $3$3	$3$3	$2\times3$2×3	$2\times2\times3$2×2×3	$2\times2\times2\times3$2×2×2×3
	Two $3$3s	$3\times3$3×3	$2\times3\times3$2×3×3	$2\times2\times3\times3$2×2×3×3	$2\times2\times2\times3\times3$2×2×2×3×3
One $5$5	No $3$3	$5$5	$2\times5$2×5	$2\times2\times5$2×2×5	$2\times2\times2\times5$2×2×2×5
	One $3$3	$3\times5$3×5	$2\times3\times5$2×3×5	$2\times2\times3\times5$2×2×3×5	$2\times2\times2\times3\times5$2×2×2×3×5
	Two $3$3s	$3\times3\times5$3×3×5	$2\times3\times3\times5$2×3×3×5	$2\times2\times3\times3\times5$2×2×3×3×5	$2\times2\times2\times3\times3\times5$2×2×2×3×3×5

This table shows all the possible ways to multiply the prime factors together. We evaluate the multiplications to find all the factors:

Factors of $360$360
		No $2$2	One $2$2	Two $2$2s	Three $2$2s
No $5$5	No $3$3	$1$1	$2$2	$4$4	$8$8
	One $3$3	$3$3	$6$6	$12$12	$24$24
	Two $3$3s	$9$9	$18$18	$36$36	$72$72
One $5$5	No $3$3	$5$5	$10$10	$20$20	$40$40
	One $3$3	$15$15	$30$30	$60$60	$120$120
	Two $3$3s	$45$45	$90$90	$180$180	$360$360

The factors of $360$360 are $1$1, $2$2, $3$3, $4$4, $5$5, $6$6, $8$8, $9$9, $10$10, $12$12, $15$15, $18$18, $20$20, $24$24, $30$30, $36$36, $40$40, $45$45, $60$60, $72$72, $90$90, $120$120, $180$180, and $360$360.

 

Practice questions

Question 1

Question 2

Question 3