Prime FactorizationWorksheets For This SkillMore SkillsFree Online Tests Trial

We call a number N a factor of another number M if N divides M. Since 1, 2, 3 and 6 all can divide 6, we say that 1, 2, 3 and 6 are factors of 6. Any number bigger than 1 has at least two factors: 1 and itself, which are called the trivial factors of that number. Prime numbers are special numbers that have only two trivial factors: 1 and itself. For example, 2 is a prime number because 2 has only two factors: 1 and 2 itself. 3 is also a prime number because 3 has only two factors: 1 and 3 itself.

A number bigger than 1 is called a composite number if it is not a prime number. 1 is neither a prime number nor a composite number. 6 is a composite number because 6 has four factors: 1, 2, 3 and 6. Among those four factors, 2 and 3 are prime numbers, and we call 2 and 3 prime factors of 6.

A number can be expressed by a product of its factors, or factored into its factors, that is called the factorization of that number. Every composite number can be factored uniquely into prime factors if the prime factors are written in order of size. For example, 6 = 2 * 3, 8 = 2 * 2 * 2, 9 = 3 * 3 and 10 = 2 * 5 , etc.

This special factorization where factors are all prime numbers is called the Prime Factorization of the composite number.

Prime Factorization is very useful. For example, if you want to find all the non-trivial factors of 24, you can first find the Prime Factorization of 24 = 2 * 2 * 2 * 3, then you can get all the non-trivial factors of 24 by taking one or more prime factors out from the Prime Factorization and calculating the product of the left-overs:
2, 3, 2 * 2 = 4, 2 * 2 * 2 = 8, 2 * 3 = 6, 2 * 2 * 3 = 12

To find out the Prime Factorization of any composite number, we need to have a list of all the prime numbers in order of size:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ...

Let us use an example to illustrate how to determine the Prime Factorization of a composite number.

Example: find the Prime Factorization of 100.
We always start with the least prime number 2. We know that 2 divides 100 and the quotient is 50. Write down:
100 = 2 * 50
Again, 2 divides 50 and the quotient is 25. Write down:
100 = 2 * 2 * 25
2 does not divide 25, the next prime number 3 does not divide 25, either. It is not hard to see that the next prime number which divides 25 is 5 and the quotient is 5. Write down:
100 = 2 * 2 * 5 * 5
Now that all the factors are already prime factors, we know the Prime Factorization of 100 is
2 * 2 * 5 * 5.

Tricks and Tips
A number is divisible by 2 if the digit in the ones place is even.
A number is divisible by 3 if the sum of the digits is divisible by 3.
A number is divisible by 5 if the digit in the ones place is 0 or 5.

Greatest Common FactorWorksheets For This SkillMore SkillsFree Online Tests Trial

The biggest factor of two or more numbers are called their Greatest Common Factor(GCF). For example, 4 is the GCF of 24 and 100 because 4 is a factor of both 24 and 100, and none of the numbers bigger than 4 is a factor of both 24 and 100.

Let us use an example to illustrate how to determine the GCF of two numbers by using the Prime Factorization of them.

Example: find the GCF of 24 and 300.

Looking at the Prime Factorization of 24 and 300,
24 = 2 * 2 * 2 * 3
300 = 2 * 2 * 3 * 5 * 5
we find that 2 appears three times in the Prime Factorization of 24 and twice in the Prime Factorization of 300, so we know that 2 * 2 is a common factor of 24 and 300, but 2 * 2 * 2 is only a factor of 24. Since 3 appears once in the Prime Factorization of 24 and 300, we know 3 is a common factor of 24 and 300. Now we know that 2 * 2 * 3 is a common factor of 24 and 300 because both 2 * 2 and 3 are common factors of 24 and 300. Since 5 * 5 only appears in the Prime Factorization of 300, we know that 5 * 5 is not a common factor of 24 and 300. In other words, we can not add more prime factors to the common factor 2 * 2 * 3. Therefore, we conclude that 2 * 2 * 3 , which turns out to be 12, is the GCF of 24 and 300.

In general, to find the GCF of two or more numbers, firstly figure out the Prime Factorization of all the numbers, then pick up the prime factors that appear in ALL the Prime Factorizations. Be sure to list each prime factor as many times as it appears in ALL the Prime Factorizations. Multiply those common prime factors together and the product is the GCF.

Reducing a Fraction to its Lowest TermsWorksheets For This SkillMore SkillsFree Online Tests Trial

A fraction can be expressed in many different ways, e.g., , , , , and , are all equivalent. In most cases, we use because the numerator and the denominator in are the smallest numbers among all the equivalent fractions of .

Reducing a fraction means finding an equivalent fraction with a smaller numerator and denominator. Reducing a fraction to its lowest terms means finding the equivalent fraction with a numerator and denominator as small as possible.

Given a fraction, to find its equivalent fraction with bigger numerator and denominator, you multiply the numerator and the denominator by the same number. To reduce a fraction, you divide the numerator and the denominator by the same number. To reduce a fraction to its lowest terms, you want to divide the numerator and the denominator by the GCF of the numerator and the denominator because after that there will be no more common factors in the numerator and the denominator.

Example: reduce to its lowest terms.
we need to find the GCF of 24 and 300, which is 12 as we found above. Since 24 divided by 12 is 2 and 300 divided by 12 is 25, reduced to its lowest terms is .

Another way to reduce is to write the Prime Factorization of the numerator on top of the Prime Factorization of the denominator, like the following:

Then take out all the common prime factors from both the top and the bottom, which are 2 * 2 * 3. What is left gives you .

Tricks and Tips
To reduce a fraction to its lowest terms, write the Prime Factorization of the numerator on top of the Prime Factorization of the denominator, then take out all the common prime factors from both the top and the bottom. The product of the prime factors left on the top is the new numerator and the product of the prime factors left at the bottom is the new denominator.

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