Edexcel GCSE Maths: Factors, Multiples, and Prime Numbers

Edexcel GCSE Maths: Factors, Multiples, and Prime Numbers

This tutorial will explore factors, multiples, and prime numbers, focusing on prime factorization, highest common factor (HCF), and lowest common multiple (LCM). You'll learn to identify prime numbers, understand divisibility rules, and apply these concepts in simplifying fractions, solving word problems, and managing large number calculations.

1. Factors and Multiples

• Factors: Numbers that divide evenly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Multiples: Numbers you get by multiplying a given number by an integer. For example, the multiples of 3 are 3, 6, 9, 12, 15, and so on.

Example:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Multiples of 7: 7, 14, 21, 28, 35, ...

2. Prime Numbers

• Prime Number: A whole number greater than 1 that has only two factors: 1 and itself.
  • Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
• Composite Number: A whole number greater than 1 that has more than two factors.
  • Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18...

Remember: 1 is neither a prime nor a composite number.

3. Prime Factorization

• Prime Factorization: Expressing a number as a product of its prime factors.
• Steps:
  1. Find a prime factor of the number.
  2. Divide the number by the prime factor.
  3. Repeat steps 1 and 2 with the quotient until you reach a prime number.
  4. Write the number as the product of all the prime factors.

Example: Prime factorization of 36

36 / 2 = 18
18 / 2 = 9
9 / 3 = 3

Therefore, the prime factorization of 36 is 2 x 2 x 3 x 3 or 2² x 3²

4. Highest Common Factor (HCF)

• HCF: The largest number that divides into two or more numbers without leaving a remainder.
• Finding HCF:
  1. Find the prime factorization of each number.
  2. Identify the common prime factors.
  3. Multiply the common prime factors, taking the lowest power of each.

Example: Find the HCF of 12 and 18.

12 = 2² x 3
18 = 2 x 3² 

Common prime factors: 2 and 3. Taking the lowest power, we get 2 x 3 = 6. Therefore, the HCF of 12 and 18 is 6.

5. Lowest Common Multiple (LCM)

• LCM: The smallest number that is a multiple of two or more numbers.
• Finding LCM:
  1. Find the prime factorization of each number.
  2. Identify all prime factors, including those not common.
  3. Multiply these prime factors, taking the highest power of each.

Example: Find the LCM of 12 and 18.

12 = 2² x 3
18 = 2 x 3² 

All prime factors: 2 and 3. Taking the highest power, we get 2² x 3² = 36. Therefore, the LCM of 12 and 18 is 36.

6. Applications

• Simplifying Fractions: Finding the HCF of the numerator and denominator helps simplify fractions to their lowest terms.
• Word Problems: HCF and LCM are used to solve problems related to sharing, grouping, and recurring events.
• Large Number Calculations: Prime factorization simplifies calculations involving large numbers, especially when dealing with division and finding square roots.

Example:

A shop sells apples in packs of 6 and oranges in packs of 8. What is the smallest number of packs you need to buy to get an equal number of apples and oranges?

Solution: We need to find the LCM of 6 and 8.

6 = 2 x 3
8 = 2³

LCM = 2³ x 3 = 24

Therefore, you need to buy 4 packs of apples (24/6) and 3 packs of oranges (24/8) to get an equal number of both.

By understanding factors, multiples, prime numbers, prime factorization, HCF, and LCM, you'll be able to tackle various problems in Edexcel GCSE Maths and beyond!