A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. This means it cannot be divided perfectly by any other number without leaving a remainder.
Example
Determine if 7 is a prime number.
solution :
We find all multiplication pairs that equal 7:
1 × 7 = 7
No other whole numbers multiply to give 7. The only factors of 7 are 1 and 7.
Result = 7 is a prime number
A composite number is a whole number greater than 1 that has more than two factors. This means it can be divided perfectly by 1, itself, and at least one other whole number.
Example
Determine if 6 is a composite number.
solution :
We find all multiplication pairs that equal 6:
1 × 6 = 6
2 × 3 = 6
The factors of 6 are 1, 2, 3, and 6. Since it has four factors, it is a composite number.
Result = 6 is a composite number
The numbers 0 and 1 are unique because they are neither prime nor composite:
Example
Classify the numbers 1, 2, and 4 as prime, composite, or neither.
solution :
The number 1 has only 1 factor, so it is neither.
The number 2 has exactly 2 factors (1 and 2), so it is prime.
The number 4 has three factors (1, 2, and 4), so it is composite.
The most famous and systematic way to find all prime numbers up to a specific limit is a method called the Sieve of Eratosthenes. This method works by progressively crossing out or filtering out multiples of prime numbers, leaving only the prime numbers behind.
Example
Find all the prime numbers between 1 and 20 using the filtering method.
solution :
Step 1: Write down all the numbers from 1 to 20.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Step 2: Filter out 1, because 1 is neither prime nor composite.
Remaining: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Step 3: Keep the first prime number, which is 2. Filter out all other multiples of 2 (all even numbers).
Numbers crossed out: 4, 6, 8, 10, 12, 14, 16, 18, 20
Remaining: 2, 3, 5, 7, 9, 11, 13, 15, 17, 19
Step 4: Keep the next prime number, which is 3. Filter out all other remaining multiples of 3.
Numbers crossed out: 9, 15
Remaining: 2, 3, 5, 7, 11, 13, 17, 19
Step 5: Keep the next prime number, which is 5. Check for remaining multiples of 5 (there are none left to cross out under 20).
Step 6: Review the final leftover list. All numbers that have not been filtered out are prime numbers.
Prime Numbers up to 20 = 2, 3, 5, 7, 11, 13, 17, 19
Prime factorization is the process of breaking down a composite number into a product of prime numbers. When you multiply these prime factors together, they equal the original number.
Repeated prime factors are often written using exponential notation (index notation) to make expressions shorter and easier to read. For example, 2 × 2 × 2 can be written as 23, where 2 is the base and 3 is the exponent (index).
This method breaks a number down into two factor branches. You continue breaking down the branches until every ending point is a prime number.
Example
Find the prime factorization of 24 using the Factor Tree method and express the answer in index notation.
solution :
Step 1: Split 24 into any two factors. Let us choose 2 and 12.
24 → 2 × 12
(Since 2 is a prime number, this branch stops here).
Step 2: Split the composite number 12 into two factors. Let us choose 2 and 6.
12 → 2 × 6
(Since 2 is a prime number, this branch stops here).
Step 3: Split the composite number 6 into two factors. These are 2 and 3.
6 → 2 × 3
(Both 2 and 3 are prime numbers, so all branches are now complete).
Step 4: Collect all the prime numbers from the ends of the branches.
Prime factors = 2 × 2 × 2 × 3
Step 5: Write the final product using exponential notation.
24 = 23 × 3
This method divides the composite number by the smallest possible prime factor. You continue dividing the resulting answers by prime numbers until you get a final answer of 1.
Example
Find the prime factorization of 60 using the Division method and express the answer in index notation.
solution :
Step 1: Divide 60 by the smallest prime factor, which is 2.
60 ÷ 2 = 30
Step 2: Divide 30 by the smallest prime factor, which is 2.
30 ÷ 2 = 15
Step 3: Divide 15 by its smallest prime factor, which is 3.
15 ÷ 3 = 5
Step 4: Divide 5 by its smallest prime factor, which is 5.
5 ÷ 5 = 1 (Since we reached 1, the division process stops here).
Step 5: Collect all the prime divisors used in the division ladder.
Prime factors = 2 × 2 × 3 × 5
Step 6: Write the final product using exponential notation.
60 = 22 × 3 × 5
Prime factorization is used to solve many different types of math problems. By breaking numbers down into their fundamental building blocks, we can easily find shared denominators, simplify expressions, and solve root values.
To find the HCF of two or more numbers, find the prime factorization of each number. Then, select the common prime factors with the lowest power and multiply them together.
Example
Find the HCF of 36 and 48 using prime factorization.
solution :
Step 1: Find the prime factorization of 36.
36 = 2 × 2 × 3 × 3 = 22 × 32
Step 2: Find the prime factorization of 48.
48 = 2 × 2 × 2 × 2 × 3 = 24 × 31
Step 3: Identify the common prime bases and select the lowest power for each.
The shared prime bases are 2 and 3.
Lowest power of base 2 is 22.
Lowest power of base 3 is 31.
Step 4: Multiply these values together.
HCF = 22 × 31 = 4 × 3 = 12
HCF = 12
To find the LCM of two or more numbers, find the prime factorization of each number. Then, list every unique prime factor present and choose the highest power for each, then multiply them together.
Example
Find the LCM of 12 and 18 using prime factorization.
solution :
Step 1: Find the prime factorization of 12.
12 = 2 × 2 × 3 = 22 × 31
Step 2: Find the prime factorization of 18.
18 = 2 × 3 × 3 = 21 × 32
Step 3: Identify all unique prime bases and select the highest power for each.
The unique prime bases are 2 and 3.
Highest power of base 2 is 22.
Highest power of base 3 is 32.
Step 4: Multiply these highest powers together.
LCM = 22 × 32 = 4 × 9 = 36
LCM = 36
To simplify a fraction to its lowest terms, find the HCF of the numerator and the denominator using prime factorization, then divide both by that HCF.
Example
Simplify the fraction 30 45 to its simplest form.
solution :
Step 1: Use prime factorization to find the components of 30 and 45.
30 = 2 × 3 × 5
45 = 3 × 3 × 5 = 32 × 5
Step 2: Find the HCF by taking the lowest power of shared prime factors.
Shared factors are 3 and 5.
HCF = 31 × 51 = 15
Step 3: Divide the numerator and the denominator by the HCF.
Numerator: 30 ÷ 15 = 2
Denominator: 45 ÷ 15 = 3
Simplest Form = 2 3
To find the square root of a perfect square, divide the exponents of its prime factors by 2. To find the cube root of a perfect cube, divide the exponents of its prime factors by 3.
Example
Find the square root of 144 using prime factorization.
solution :
Step 1: Find the prime factorization of 144.
144 = 2 × 2 × 2 × 2 × 3 × 3 = 24 × 32
Step 2: Divide each exponent by 2 to find the square root.
Exponent of base 2 becomes: 4 ÷ 2 = 2
Exponent of base 3 becomes: 2 ÷ 2 = 1
Step 3: Multiply the simplified prime bases together.
Square Root = 22 × 31 = 4 × 3 = 12
Square Root of 144 = 12
Example 1
Find the prime factorization of 90 and express the answer in index notation.
solution :
Step 1: Divide 90 by its smallest prime factor, which is 2.
90 ÷ 2 = 45
Step 2: Divide 45 by its smallest prime factor, which is 3.
45 ÷ 3 = 15
Step 3: Divide 15 by its smallest prime factor, which is 3.
15 ÷ 3 = 5
Step 4: Divide 5 by its smallest prime factor, which is 5.
5 ÷ 5 = 1 (The process stops here).
Step 5: Collect the prime divisors and apply index notation for repeated factors.
Prime factors = 2 × 3 × 3 × 5 = 2 × 32 × 5
Final Answer = 2 × 32 × 5
Example 2
Find the Highest Common Factor (HCF) of 72 and 120 using the prime factorization method.
solution :
Step 1: Find the prime factorization of 72.
72 = 2 × 2 × 2 × 3 × 3 = 23 × 32
Step 2: Find the prime factorization of 120.
120 = 2 × 2 × 2 × 3 × 5 = 23 × 31 × 51
Step 3: Identify the shared prime bases and choose the lowest power for each.
Shared prime bases are 2 and 3.
Lowest power of base 2 is 23.
Lowest power of base 3 is 31.
Step 4: Multiply these choices together to get the HCF.
HCF = 23 × 31 = 8 × 3 = 24
Final Answer = 24
Example 3
Find the Lowest Common Multiple (LCM) of 20 and 35 using prime factorization.
solution :
Step 1: Find the prime factorization of 20.
20 = 2 × 2 × 5 = 22 × 51
Step 2: Find the prime factorization of 35.
35 = 5 × 7 = 51 × 71
Step 3: Identify all unique prime bases and choose the highest power for each.
The unique prime bases present are 2, 5, and 7.
Highest power of base 2 is 22.
Highest power of base 5 is 51.
Highest power of base 7 is 71.
Step 4: Multiply these highest powers together to get the LCM.
LCM = 22 × 51 × 71 = 4 × 5 × 7 = 140
Final Answer = 140
Example 4
Find the cube root of 216 using the prime factorization method.
solution :
Step 1: Find the prime factorization of 216.
216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33
Step 2: Divide the exponent of each prime factor by 3 to find the cube root.
Exponent of base 2 becomes: 3 ÷ 3 = 1
Exponent of base 3 becomes: 3 ÷ 3 = 1
Step 3: Multiply the simplified prime factors together.
Cube Root = 21 × 31 = 2 × 3 = 6
Final Answer = 6
Example 5
Reduce the fraction 84 96 to its simplest form using prime factors.
solution :
Step 1: List the prime factor expressions for the numerator and denominator.
84 = 2 × 2 × 3 × 7 = 22 × 31 × 71
96 = 2 × 2 × 2 × 2 × 2 × 3 = 25 × 31
Step 2: Find the HCF of 84 and 96 by choosing the lowest power of shared prime bases.
Shared bases are 2 and 3.
HCF = 22 × 31 = 4 × 3 = 12
Step 3: Divide the numerator and denominator by the HCF (12).
Numerator: 84 ÷ 12 = 7
Denominator: 96 ÷ 12 = 8
Final Answer = 7 8
Attempt the following Questions:
(1.) Express 180 as a product of prime factors in index notation.
(2.) Find the Highest Common Factor (HCF) of 54 and 90 using prime factorization.
(3.) Find the Lowest Common Multiple (LCM) of 24 and 30 using prime factorization.
(4.) Find the square root of 400 using the prime factorization method.
(5.) Reduce the fraction 60 72 to its simplest form using prime factors.
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