Download GCF and LCM by Prime Factorization

GCF and LCM by Prime Factorization
I. To Find the GCF
GCF stands for Greatest Common Factor. Factors are numbers multiplied together. Sometimes,
it’s necessary to know the GCF of 2 or more numbers. Why? I’m glad you asked. Basically, the
GCF helps to efficiently and quickly reduce fractions. One way to determine the GCF is simply
by prime factorization.
When to use : to reduce fractions
Skills needed: multiplication facts, prime factorization
Example: 12 & 36 Find the GCF
Step 1: Find the Prime factors for
12: 2 x 2 x 3
36: 2 x 2 x 3 x 3
Step 2: Multiply each factor that has a match
GCF = 2 x 2 x 3 = 12
Example: Find the GCF for 48 and 72
Step 1: Find the prime factors for
48: 2 x 2 x 2 x 2 x 3
72: 2 x 2 x 2 x 3 x 3
Step 2: Multiply each factor that has a match
GCF = 2 x 2 x 2 x 3 = 24
Example: Find the GCF for 56, 70, and 98
Step 1: Find the prime factors for
56: 2 x 2 x 2 x 7
70: 2 x 5 x 7
98: 2 x 7 x 7
Step 2: Multiply each factor that has a match in all 3 numbers
GCF = 2 x 7 = 14
Application
Reduce or simplify the fraction, 48/72.
1. Find the GCF (We know that the GCF for 48 & 72 is 24)
2. Divide the numerator and denominator by 24. (48/24 and 72/24)
3. Thus 48/72 = 2/3
II. To Find the LCM
When to use:when adding or subtracting unlike denominators, the LCM is needed to find the
least common denominator
Skills needed: multiplication facts, prime factorization
LCM stands for Least Common Multiple (LCM). - Multiples are the result of repeated addition or
what you referred to as skip counting in Elementary school
For example, the multiples of 2 are below
2 >>> 2, 4, 6, 8, 12, 14, 16, 18, 24 . . .
In the above section, we found the GCF. Let’s use the first two examples, but find the LCM.
Example: Find the LCM of 12 and 36
Step 1: Find the Prime factors for
12 = 2 x 2 x 3
36 = 2 x 2 x 3 x 3
Step 2: Multiply each prime factor the greatest number of times it appears in any one
factorization.
The most number of times 2 appears in either factorization is twice. The most number of times 3
appears in either factorization is twice. Thus,
LCM = 2 x 2 x 3 x 3 = 36
Example: Find the LCM for 48 and 72
Step 1: Find the prime factors for
48: 2 x 2 x 2 x 2 x 3
72: 2 x 2 x 2 x 3 x 3
Step 2: Multiply each prime factor the greatest number of times it appears in any one
factorization. The most number of times 2 appears in either factorization is four times. The most
number of times 3 appears in either factorization is twice. Thus,
LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144
Example: Find the LCM for 15, 25, and 8
Step 1: Find the prime factors for
6=2x3
15 = 3 x 5
25 = 5 x 5
Step 2: Multiply each prime factor the greatest number of times it appears in any one
factorization. The most number of times 2 appears in either factorization is once. The most
number of times 3 appears in either factorization is once. The most number of times 5 appears in
either factorization is twice. Thus,
LCM = 2 x 3 x 5 x 5 = 150
Application:
Add 5/12 and 7/36
The denominators are not the same. So, find the LCM, also referred to as the least common
denominator (LCD). The LCD is 36. (Refer to example above)
The equivalent fraction for 5/12 using 36 as the denominator is 15/36. 7/36 stays the same.
Therefore 15/36 + 7/36 = 22/36.
Reduce or simplify 22/36 to its lowest terms. Use the greatest common factor (GCF).
The GCF for 22 and 36 is 2.
Divide both 22 and 36 by 2. 22/36 = 11/18
Summary: 5/12 + 7/36 = 15/36 + 7/36 = 22/36 = 11/18
Divisibility Rules
2
3
4
5
6
7
8
9
10
If the last digit is even, the number is divisible by 2.
If the sum of the digits is divisible by 3, the number
is also.
If the last two digits form a number divisible by 4,
the number is also.
If the last digit is a 5 or a 0, the number is divisible
by 5.
If the number is divisible by both 3 and 2, it is also
divisible by 6.
Take the last digit, double it, and subtract it from the
rest of the number; if the answer is divisible by 7
(including 0), then the number is also.
If the last three digits form a number divisible by 8,
then the whole number is also divisible by 8.
If the sum of the digits is divisible by 9, the number
is also.
If the number ends in 0, it is divisible by 10.