Prime Factor Decomposition | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Prime Factor Decomposition

What are prime factors? 

• Factors are numbers that are multiplied together
• Prime numbers are numbers that have exactly two factors - themselves and 1
• The prime factors of a number are all the prime numbers that multiply to give that number
• It's important to remember the initial prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, ...

How do I find the prime factors?

• Utilize a factor tree to identify prime factors of a number.
  • Decompose a number into a pair of factors that, when multiplied together, yield the original number.
  • Repeat the process of splitting numbers until reaching a prime number.
    • Prime numbers cannot be divided into factors other than 1 and themselves.
• A number can be expressed uniquely as a product of its prime factors.
  • Write the prime factors in ascending order with '×' between them, for example: 72 = 2 × 2 × 2 × 3 × 3.
  • For clarity, express using powers if requested, like: 72 = 2³ × 3².

How might a question be worded?

Indeed, this topic allows for questions to employ various phrases that convey equivalent meanings.

• Express … as the product of prime factors
• Find the prime factor decomposition of …
• Find the prime factorisation of …

Uses of Prime Factor Decomposition

Prime factor decomposition (PFD) is a method used to express a number as a product of its prime factors. It has various applications in mathematics, such as determining whether a number is a square or a cube number, and even finding the square root of a number without using a calculator.

How can I use PFD to tell if a number is a square or a cube number?

Square Numbers

• To determine if a number is a square number using its prime factor decomposition, check if all the exponents of the prime factors are even. If they are all even, then the number is a square number.
  • For example, the prime factorization of 7056 is 2⁴ × 3² × 7², indicating that it is a square number.

Cube Numbers

• For determining cube numbers through prime factor decomposition, observe if all the exponents of the prime factors are multiples of 3. If this condition is met, the number is a cube number.
  • As an illustration, the prime factorization of 1728000 is 2⁹ × 3³ × 5³, indicating that it is a cube number.

How can I use PFD to find the square root of a square number?

• Start by writing the number in its prime factor decomposition.
  • Ensure that all the indices are even if it is a square number.
• Halve all of the indices present.
• This new set of factors is the prime factor decomposition of the square root of the original number.
  • If you need to express the square root as an integer, multiply the prime factors together.

How can I use PFD to find the exact square root of a number?

To find the square root of a number expressed as a product of its prime factors, follow these steps:

Example:

• For instance, consider the number 1728000. Its prime factor decomposition is 2^9 × 3^3 × 5^3, indicating that it is a cube number.

Finding the Exact Square Root of a Number using PFD

Here are the steps to find the square root of a number written as a product of its prime factors:

• Step 1: Write out the prime factors as individual factors.
• Step 2: Pair the factors together so that any identical prime factors can be written once as a power of 2.
• Step 3: Calculate the product of each of these paired prime factors, ignoring that each one is squared. This product will be written as an integer in front of the square root symbol.
• Step 4: Multiply the remaining factors together. Ensure that none of the remaining factors are the same.
• Step 5: Express the answer as a product of the integer from Step 3 and the square root of the integer from Step 4.
• For example, consider the prime factor decomposition of 360 as 2³ × 3² × 5
  • This can be written as 2² × 2 × 3² × 5 or 2²  × 3² × 2 × 5
  • So the exact square root of 360 is