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Factors
General Meaning of Factor
So, since 2 x 3 = 6, we say that “2 is a factor of 6” and “3 is a
factor of 6”.
Another way to think of factors is this:
Any number that “goes into” 18 is called a “factor of 18”.
Note that 2 goes into 18 (9 times with no remainder), so we say
that “2 is a factor of 18”.
Note that 5 does NOT go into 18 evenly (gives remainder 3), and
thus 5 IS NOT a factor or 18.
So in general, a factor of some number “x” must go into “x” evenly
without any remainder.
Example 1: Is 8 a factor of 12?
Solution: 8 goes into 12, once remainder 4. So NO. It is
not.
Example 2: Is 7 a factor of 42?
Solution: 7 goes into 42, 6 times no remainder. So Yes.
It is. And we see that 6 x 7 = 42. Thus both 6 and 7 are
factors of 42.
CAUTION: It is important to realize that saying “7 is a factor” is
MEANINGLESS. You can only be a factor “of a number”. So 7 is a
factor of 14, and a factor of 21, and a factor of 28 etc., but you
can’t just say “7 is a factor”.
List of All Factors of a Number
When trying to find factors of 12, you may find:
If you check all the other numbers (5,7,8,9,10,11) you will see that none of these
are factors of 12. Thus, [1,2,3,4,6,and 12] makes a list of ALL THE FACTORS of
12. A list of factors can be found for all other numbers too.
How do you find the list of all factors of a number?
Find 2 factors at a time, working from the outside in. When you cross over, you
know you’ve covered everything…. check it out
Example: All the factors of 20.
Start at 1: 1×20=20, so put 1 at the start, and put its "partner" 20 at the other
end:
1
20
Then go to 2. 2×10=20, so put in 2 and 10:
1
2
10
20
Then go to 3. 3 doesn't work (3×6=18, 3×7=21). 3 does not go into 20.
Then on to 4. 4×5=20, so put them in:
1
2
4
5
10
20
There is no whole number between 4 and 5 so you are done!
1
2
4
5
10
20
Factor Pairs
You may have noticed that factors of a number come in pairs. For factors of 20,
1x20, 2x10, and 4x5 all made 20. Note each factor has a partner.
If you are trying to find a factor pair of 12, you need to find 2 numbers that
multiply to give 12.
Since 3x4 = 12, 3 and 4 are a factor pair for 12.
Example:
A list of all the factors of 24 is [1,2,3,4,6,8,12,24]. Find all
the factor pairs from this.
Solution:
Ok, so 1 x 2 = 2, NOT 24, so they aren’t a pair. A better
way would be to pair them up from the outside and work in.
Note, 1 x 24 = 24, so 1 and 24 are a pair.
2 x 12 = 24, so they too are a pair.
3 x 8 = 24, so they are a pair.
4 x 6 = 24, so that must be the final pair.
NOTE: For factors of 12, 3 and 4 were a pair. But for 24, they were NOT a pair,
even though they were both factors. Factor Pairs for a certain number must
multiply together to make that number.
A Trick with Factors
When working with really big numbers, a trick people use to find NEW factor pairs
from old ones is the DOUBLE / HALF or equivalent trick.
Note that 5 x 8 = 40. So 5 and 8 are a factor pair. Now, if I double 5, but half 8, I
won’t change the product because doubling and halving are opposites and cancel
each other out. So I get 10 x 4 = 40.
Other opposites that cancel out are third/triple, quarter/times by 4,
divide by 5/times by 5, divide by 6/times by 6 etc. etc. Hopefully you have seen
the pattern.
Factor Pairs with Square Numbers
When making a list of factors, the total number of factors is usually even. The
only time this isn’t the case is with square numbers, like 25 and 36. Let’s have a
look why.
Example: All the factors of 36.
Start at 1: 1×36=36, so put 1 at the start, and put its "partner" 36 at the other
end:
1
36
Then go to 2. 2×18=36, so put in 2 and 18:
1
2
18
36
10
20
Then on to 3. 3×12=36, so put them in:
1
2
3
12
Then on to 4. 4×9=36, so put them in:
1
2
3
4 9 12
10
20
Then go to 5. 5 doesn't work (5×7=35, 5×8=40). 5 does not go into 36.
Then on to 6. 6×6=36, So 6 pairs with itself. But we will only put it once in the
list. Thus we get an odd number of factors.
1
2
3
4 6 9 12
10
20
This will only happen with square numbers, where some number multiplied by
itself will be a factor pair and only count as 1 factor in the list.
Applications with Factors
Factors are useful when we are trying to group things evenly. For example, if a
teacher has 24 students, we can use a list of all the factors of 24 to work out how
he or she could group his/her students.
So… the factors of 24 are 1,2,3,4,6,8,12,24.
So the teacher could have 24 groups of 1 (they’re not really groups then though!)
… or 12 groups of 2s (pairs)
…or 8 groups of 3
… or 6 groups of 4
… or 4 groups of 6
… or 3 groups of 8
… or 2 groups of 12
… or 1 group of 24 (that’s just the whole class!).
What’s my age Questions
Often puzzles can use factors as hints. For example, in a “What’s my age”
question, the clues could be
1) I’m less than 70
2) 4 is a factor of my age
3) 6 is a factor of my age
4) 5 is a factor of my age
A strategy is to always make a list of all the numbers that the LARGEST factor
mention goes into. So here we find all the numbers that 6 goes into, in other
words, all the multiples of 6.
So based purely on CLUE 3), the answer could be:
6
12
18
24
30
36
42
48
54
60
66
Notice I stopped at 66 because of CLUE 1 (less than 70).
From here we use the next biggest factor to cross some things off our list. In
order to satisfy CLUE 4, we need the number to be in the 5 times tables, or else 5
wouldn’t be a factor.
So I can cross off all the numbers except 30 and 60.
6
12
18
24
30
36
42
48
54
60
66
Then, finally using CLUE 2, I need the number to be divisible by 4 too. 30 is NOT
divisible by 4, so 4 wouldn’t be a factor of 30. But it is of 60. So the answer must
be 60.
NOTE: I could have used the clues in any order I liked. The reason I started with
6 is a factor was it made by original line-up of possible answers smaller. If I did 4
is a factor, I would have had to do all my 4 times tables up to 70 and this would
include a lot more numbers.
www.avilamaths.com/secret.php is a secret page in Avila Maths (not so secret
now) that will give you a list of all the factors in a number when you enter a
number and then click the button. This may be useful for checking your answers.