WHOLE NUMBERS-FACTORS & FACTORING

• Definitions related to factoring
• Prime numbers & composites
• Factoring
• Factor trees-Breaking a number down to it's prime roots
• Go to the Whole number home page
• contact me

3 divdes 6 since 6/3 = 2 with no remainder.
However, 3 does not divide 7 since 7/3 = 2 with 1 remainder.

12 can be divided by 1, 2, 3, 4, 6, and 12. therefore each of these numbers {1,2,3,4,6,12} can be called factors of 12
Similarly 10 divides 100 therefore 10 is a factor of 100
However, 10 does not divide 22, therefore 10 is not a factor of 22

12 is a multiple of 4, since 4 * 3 = 12. (12 is also a multiple of 3)
Other multiples of 4 include 8, 16, 20, and 24.
35 is a multiple of 5, since 5 * 7 = 35. (35 is also a multiple of 7)
Other multiples of 5 include 10, 15, 20, 25, and 30.

11 is a prime number since 11 divides 11, 1 divides 11, but no other number divides 11.
21 is a composite number since it can be divides by 3 and 7 as well as 1 and itself.
23 is a prime number since 23 divides 23, 1 divides 23, but no other number divides 23.
122 is a composite number since it can be divides by 2 and 61 (as well as other factors) as well as 1 and itself.
5 is a prime number since 5 divides 5, 1 divides 5, but no other number divides 5
81 is a composite number since it can be divides by 9 as well as 1 and itself.
19 is a prime number since 19 divides itself, 1 divides it, but no other number divides it
6 is a composite number since it can be divides by 3 and 2 as well as 1 and itself.
2 is prime, since it can only be divided by itself and 1

In fact 2 is the only even prime number, since all other even numbers can be divided by themselves, 1, and 2. All other even numbers are composite.

Pay attention as I find the set of factors for 96

Starting with 1. 96/1 = 96. Our first two factors are 1 and 96
Going on to 2. 96/2 = 48. Now our factors are {1,2,48,96}
3 does not divide 96
96/4 = 24. The factors are now {1,2,4,24,48,96}
5 does not divide 96
96/6=16. The factors are now {1,2,4,6,16,24,48,96}
7 does not divide 96
96/8=12. The factors are now {1,2,4,6,8,12,16,24,48,96}
9 does not divide 96
10 does not divide 96
11 does not divide 96
Which brings us to 12. We already encountered 12, so our list is now complete.

The factors of 96 are {1,2,4,6,8,12,16,24,48,96}

Find the set of factors for 36

Starting with 1. 36/1 = 36. Our first two factors are 1 and 36
36/2 = 18. The factors are now {1,2,18,36}
36/3 = 12. The factors are now {1,2,3,12,18,36}
36/4 = 9. The factors are now {1,2,3,4,9,12,18,36}
5 does not divide 36
36/6 = 6. Now the factors are 6 and 6. We need only list "6" once in our list of factors AND we have repeated a number, so our list is complete The factors of 36 are The factors are now {1,2,3,4,6,9,12,18,36}

There are several techniques to do this, but I have found the easiest way involves using trees (no, not the kind you plant).
Pick any two factors for your number, and write down both of the factors, connecting them to the original number with a line. These factors are now known as branches.
Now branch our these two factors to two more factors
And branch out the facotrs for those factors
Keep going until all that is left is a list of prime numbers
Now read off the prime numbers at the end of each branch and you'll have the prime representation of your original number
(i.e. If you multiplied all of these prime numbers together you would get your original number).

What is the prime representation for 56?
Now I know 2 divides 56. 56/2=28. So 2 and 28 are 2 factors of 56. Lets start by creating two branches for 56, one for 2 and one for 28.

Now I know 2 is prime, so that branch cannot go any further. However, 28 can be further divided. I know 2 divides 28 and 28/2=14, so lets create two branches for 28, one for 2 and one for 14.

Now I know 2 is prime, so that branch cannot go any further, but 14 can still be further divided. Now 2 divides 14 and 14/2=7, so lets create two more branches.

The last two branches are 2 and 7. Both of these numbers are prime and so the tree cannot go any further.

Now read the numbers at the end of each branch for the prime representation of 56 (the ones marked in red) and we see that 56=2*2*2*7

This time I'm going to map out 150.
Now just to name two factors, I know 150=15*10.


Now this time both 15 and 10 can be subdivided, so both branches of the tree can be continued. 15=5*3 and 10=2*5

Now our branches end with 5, 3, 2, and 5. all of these numbers are prime, so this is the end of the tree. 150=2*3*5*5