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Ch. 5.1: Factors, Factoring Integers, Finding the GCF
Transcript of Ch. 5.1: Factors, Factoring Integers, Finding the GCF
Factors will play a crucial role in algebra, as you will see. Having the skills to factor numbers into factor pairs and into prime factors, and finding GCFs and LCMs will make many complex algebra tasks much simpler. It's important to master factoring skills early on to succeed in algebra!
Factors, Factoring Integers, Finding Greatest Common Factors and Least Common Multiples
Two or more numbers that can be multiplied together to get the original number.
Factoring a Number
There are two useful ways to factor integers:
1) finding "factor pairs", and
2) finding the "prime factorization".
What are factors? How are factors of integers calculated? How are Greatest Common Factors and Least Common Multiples found?
1 and 10
-1 and -10
2 and 5
-2 and -5
1 and 30
2 and 15
3 and 10
5 and 6
2, 3, and 5
To find factor pairs of a number, simply divide the number by some other number.
To find the prime factorization of a number, simply divide it by a prime number (except 1), then divide the result by another prime number. Keep repeating until all the factors are prime.
To find the factor pairs of 12 (other than 1 and 12!), we could divide 12 by 2 to get the factor pair 2 and 6 (and don't forget -2 and -6! From now on in the presentation, I'll rely on you to "remember the negatives").
We could also divide 12 by 3 to get the factor pair of 3 and 4.
The prime factorization of 12 is:
The prime factorization of 28 is:
The prime factorization of 225 is:
Besides 1 and itself, 28 can be factored into the factor pairs of 2 and 14, and 4 and 7.
factors are numbers multiplied together to form the original number;
find factors by dividing the original number;
remember the negatives;
there are two useful ways to factor integers: factor pairs, and prime factorization;
GCFs and LCMs can be found by prime factorization of each number.
Integers are the set of whole numbers and their opposites. Or, to think of it another way, 0 and all the positive and negative numbers without decimals: ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, and so on.
-1 and 15
1 and -15
-3 and 5
3 and -5
... as well as combinations of their opposites/negatives.
If the number needing to be factored is negative, simply factor out the -1 first:
-1 * 2 * 3 * 7
Greatest Common Factors
The greatest common factor (or "GCF") of several numbers is the biggest number that each of the original numbers is divisible by. This is just another way of saying "the biggest number that is a factor of all of them": the *greatest common factor*!
To find the GCF:
first, factor each of the original numbers down to primes;
next, borrow each base factor that appears in *all* of them (ignore others);
now borrow the smallest power of those base factors;
lastly, multiply the result out to get the GCF.
Find the GCF of 72 and 540.
by Patrick Bennett, Saint Mary's High School
Least Common Multiples / Least Common Denominators
As we just saw, the Greatest Common Factor of several numbers is the biggest number that each of the original numbers is divisible by. On the other hand, the Least Common Multiple of several numbers is the smallest number that is divisible by all of the original numbers. Least Common Multiples are used often when working with the bottoms of fractions (the "denominators"), so another term used for them a lot is "Least Common Denominator" or "LCD". They are basically the same thing.
Finding the LCM/LCD:
Factor each number to primes;
Borrow each base factor, along with it's greatest power;
Multiply out to find the LCM/LCD.
Greatest Common Factor ("GCF"): product of the *smaller* powers of each *common* prime factor.
Least Common Multiple ("LCM"/"LCD"): product of the *larger* powers of *each* prime factor.
A lot of the time it's easy to see what the LCM is. Other times it isn't so easy, and we'll need a reliable method for finding it - this is where factoring again comes to the rescue!
Eg. Find the LCM of 9, 15, and 24.