Prime Factorization and Factoring
Using the “Ladder” Method
There are several levels to the Ladder.
• Here are all the prime numbers less than 20
2, 3, 5, 7, 11, 13, 17, 19
•
You will begin factoring using the lowest prime number
(2), then work their way through the next primes until
they reach a prime number at the bottom
Factoring using The Ladder Method
Factoring of one number:
The number to factor is 24
•
•
Ask: will 2 go into 24?.... Yes.
How many times? (12)
12
•
•
Will 2 go into 12?..... Yes.
How many times? (6)
6
•
•
Will 2 go into 6?.... Yes
How many times? (3)
2
24
2
2
3
Another Example:
The number to factor is 52.
2
52
2
26
13
•
•
Student asks: will 2 go into 52?.... Yes.
How many times? (26)
•
•
Will 2 go into 26?.... Yes.
How many times? (13)
•
•
•
Will 2 go into 13?.... No.
Will 3 go into 13?.... No.
Recognize that 13 is prime so it cannot be
factored any more
So, the Prime Factorization is:
2 × 2 × 13
or
So, the Prime Factorization is:
2× 2× 2×3
or
23 × 3
Notice how nicely and neatly the factors are
lined up. It is more organized than the usual
factor tree method.
-2-
22 ×13
XMG Playbook ©
How to factor a numerator and a denominator for
simplifying fractions: Use the Ladder to pull out
the common prime factors.
You can use the Ladder to find the Greatest common
Factor(GCF) and the Least Common Multiple(LCM).
The fraction to simplify is:
•
•
2
24
36
Make a “double Ladder” in order to factor out the
common factor.
First Label the numerator and the denominator.
Ask:
N
D
24
36
• Will 2 go into 24 and 36?..... Yes.
• How many times for each?
• Will 2 go into 12 and 18?.... Yes.
• How many times for each?
2
12
18
3
6
9
2/ 3
Finding the GCF and LCM
Using the ladder you can find the Greatest Common Factor GCF
and the Least Common Multiple LCM:
GCF is 2 × 2 × 3 = 12
(Notice how these factors run along the
side of the ladder and when multiplied,
the result is the greatest common
factor.)
LCM is 2 × 2 × 3 × 2 × 3 = 72
(Notice how these factors form the
shape of an “L” (for Ladder or LCM) and
when multiplied, the result is the Least
Common Multiple.)
The fractions to add are:
•
• Will 2 go into 12 and 18?…. No.
• Will 3 go into 12 and 18?…. Yes.
• How many times for each? 3
In using the ladder, you pull out the common factors leaving
the simplified form (numerator and denominator) on the
bottom rung of the ladder.
So,
24 2
=
36 3
As you “grow up” in factoring. You can take out a larger
factor if you can recognize it.
For example, on the first rung, if you saw that 6 was a factor
of each number you could start with that number. Then you
would have to factor 4 and 6 on the next rung.
N
24
D
36
2
12
18
3
6
9
2
3
1 2
+
8 6
Make a “double Ladder” in order to factor out the least
common multiple for the denominators
Put the denominators into the ladder
•
Ask:
2
8
6
4
3
•
•
•
•
•
Will 2 go into 8 and 6?..... Yes
How many times for each?
Will 2 go into 4 and 3?…. No
Will 3 go into 4 and 3?…. No
no more factoring can be done.
Remember, the factors along the outside of the ladder make the
Least Common Multiple when multiplied with each other.
So, 2 x3 x 4= 24. The LCM is 24.
Even better though is how this ladder method sets apart the
factors for each denominator.
The bottom rung leaves factors which are not common between
the two denominators. Now, the student just has to multiply the
two fractions by the missing factor for each denominator.
Example:
1(3) 2(4) 3
8
+
=
+
8(3) 6(4) 24 24
3
nmcguire, 2005
2
-4-
which is
11
24