A factor is a number that divides another number without producing a remainder. To put it in a related manner, if multiplying two whole numbers yields a product, the numbers we are multiplying are factors of the product because they are divisible by it.
For example, 2 × 3 = 6. As a result, 2 and 3 are both factors of 6. When 6 is divided by either 2 or 3, there is no remainder.
A factor pair is a set of two factors that, when multiplied, produce a specific product.
Example: Calculate the factor pairs of 24.
Factors of 24: 1, 2, 3, 4, 6, 8, 12 and 24
Factor pairs of 24: ( 1, 24 ), ( 3, 8 ), ( 2, 12 ), ( 4, 6 )
Numbers that are greater than one and have no other factors other than 1 and itself are known as prime numbers. They only have one and the number itself as factors. This also indicates that these numbers cannot be divided by anything other than 1 and the number itself without a remainder.
For example, 7 = 1 × 7, where 7 is a prime number because it contains only two factors, that is, 1 and itself.
Composite numbers are whole numbers with more than two factors. In other words, composite numbers are whole numbers that are not prime and are divisible by more than two numbers.
Prime factorization is a method of writing a number as a product of its prime factors. A composite number has more than two factors, hence this method is suitable for composite numbers to express all its prime factors.
The prime factorization of 252 is:
252 = 2 x 2 x 3 x 3 x 7 = \(2^2\) x \(3^2\) x 7
2, 3, 7 are prime numbers. Hence, the prime factors of 252 will be 2, 3 and 7.
The factor tree method involves finding the factors of a number and then factorizing those numbers until we reach the prime factors. The steps below are used to determine the prime factorization of a number using the factor tree method.
Step 1: Place the number at the very top of the factor tree.
Step 2: Then, write down the corresponding factor pairs in the form of branches of the factor tree.
Step 3: Factorize the composite factors found in step 2 and write the factor pairs of the number below as the next branches of the tree.
Step 4: The above step is repeated until all of the composite numbers are expressed as products of prime factors.
Example: What are the two-factor trees for the number 60?
We can make two different factor trees because 60 is a composite number.
First factor tree:
60 = 30 x 2 (Factor pair of 60)
30 = 6 x 5 (Factor pair of 30)
6 = 2 x 3 (Factor pair of 6)
Second factor tree:
60 = 15 x 4 (Factor pair of 60)
15 = 3 x 5 (Factor pair of 15)
4 = 2 x 2 (Factor pair of 4)
The prime factors of 60 are the same in both factor trees, which are 2, 2, 3, and 5.
Prime factorization of 60: 2 x 2 x 3 x 5 = \(2^2\) x 3 x 5
A perfect square is an integer that can be expressed as the square of another integer. In other words, it is defined as the product of an integer when it is multiplied by itself.
Let us consider an example to understand the method we can apply to find the greatest perfect square that is a factor of a number using prime factorization.
What is the greatest perfect square that is a factor of 90?
Write the prime factorization of 90 using a factor tree. Then look for perfect square factors.
The prime factorization clearly shows that 90 has one perfect square factor and that is, 3 ⋅ 3 = 9
As a result, the largest perfect square with a factor of 90 is 9.
Calculate the factor pairs of 150.
As a result, the factor pairs of 150 are ( 1 , 150 ), ( 2 , 75 ), ( 3 , 50 ), ( 5 , 30 ), ( 6, 25 ), and ( 10 , 15 ).
Write the prime factorization and find the prime factors of 850 using a factor tree.
Hence, the prime factorization of 850 is 2 x 5 x 5 x 17 = 2 x \(5^2\) x 17
What is the greatest perfect square that is a factor of 72?
Write the prime factorization of 72 using a factor tree. Then look for perfect square factors.
The prime factorization of 72 is 2 x 2 x 2 x 3 x 3 = \(2^3\) x \(3^2\)
3 ⋅ 3 = 9
2 . 2 = 4
( 2 . 3 ) ⋅ ( 2 . 3 ) = 6 . 6 = 36
As a result, the largest perfect square, which is a factor of 72 is 36.
The “dot” symbol represents multiplication. Multiplication is represented by and also by a dot between two numbers. For example, “3 ⋅ 4 = 12”
A prime number is a number that has 2 factors, 1 and itself. Composite numbers, on the other hand, are numbers that have more than 2 factors. 1, on the other hand, has only 1 factor, which is 1 itself. Hence, 1 is neither prime nor composite.
A perfect number is a positive integer that is equal to the sum of its divisors, excluding the number itself.
For example, 6 is the smallest perfect number and the factors of 6 are 1, 2, and 3. The sum of 1 + 2 + 3 = 6. Hence, 6 is a perfect number.