Ever wondered how to share items in equal groups, or when two events will next happen at the same time? The maths behind these questions involves understanding factors, multiples, and prime numbers.
📅 Scheduling & Planning
Figuring out when one bus arriving every 12 minutes and another every 18 minutes will arrive together uses multiples.
🍰 Fair Sharing
If you have 24 sweets to share equally, knowing how many friends you can invite involves finding the factors of 24.
🔒 Cybersecurity
The encryption that keeps your online banking safe relies on the special properties of very large prime numbers.
Understanding the Core Concepts
Factors (or Divisors)
A factor is a whole number that divides into another number exactly, with no remainder. Factors of 12 are 1, 2, 3, 4, 6, and 12. They often come in pairs: (1, 12), (2, 6), and (3, 4).
Multiples
A multiple of a number is what you get when you multiply it by any whole number. Think of it as a number’s “times table”. The first few multiples of 5 are 5, 10, 15, 20, and so on.
Prime Numbers
A prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The number 1 is not prime. The first few primes are 2, 3, 5, 7, 11, 13… Remember, 2 is the only even prime number!
Finding What’s in Common
Highest Common Factor (HCF)
The HCF is the largest factor that two or more numbers share. For 12 {1,2,3,4,6,12} and 18 {1,2,3,6,9,18}, the HCF is 6.
Lowest Common Multiple (LCM)
The LCM is the smallest multiple that two or more numbers share. For 4 {4,8,12,16…} and 6 {6,12,18…}, the LCM is 12.
Prime Factorisation
This is breaking a number down into a product of its prime factors. A factor tree is a great way to do this. The prime factorisation of 72 is $2 \times 2 \times 2 \times 3 \times 3$, written in index notation as $72 = 2^3 \times 3^2$.
Worked Examples
Example 1: Finding HCF by Listing
Find the HCF of 24 and 40.
- Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}.
- Factors of 40: {1, 2, 4, 5, 8, 10, 20, 40}.
- The highest common factor is 8.
Example 2: Finding LCM by Listing
Find the LCM of 9 and 15.
- Multiples of 9: {9, 18, 27, 36, 45, …}.
- Multiples of 15: {15, 30, 45, …}.
- The lowest common multiple is 45.
Example 3: HCF with Prime Factors
Find the HCF of 48 and 72.
- Prime factors of 48 $= 2^4 \times 3$.
- Prime factors of 72 $= 2^3 \times 3^2$.
- Take the common primes with the lowest powers: $2^3$ and $3^1$.
HCF $= 2^3 \times 3 = 8 \times 3 = \mathbf{24}$.
Example 4: LCM with Prime Factors
Find the LCM of 45 and 60.
- Prime factors of 45 $= 3^2 \times 5$.
- Prime factors of 60 $= 2^2 \times 3 \times 5$.
- Take all primes with the highest powers: $2^2$, $3^2$, and $5^1$.
LCM $= 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = \mathbf{180}$.
Practice Questions
- List all the factors of 36.
- Which of these are prime? 1, 7, 15, 23, 29, 39.
- Find the LCM of 9 and 15 by listing their multiples.
- Write 90 as a product of its prime factors, using index notation.
- A baker has 54 chocolate biscuits and 42 shortbread biscuits. What is the greatest number of identical bags she can make? (Hint: HCF or LCM?)
Show Answers
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
- Prime numbers: 7, 23, 29. (1 is not prime; 15 = 3×5; 39 = 3×13.)
- LCM(9, 15) = 45. (Multiples of 9: 9, 18, 27, 36, 45…; Multiples of 15: 15, 30, 45…)
- $90 = 2 \times 3^2 \times 5$.
- This is an HCF problem (“greatest number of identical bags”). HCF(54, 42) = 6 bags.
FAQs
Q: What’s the real difference between a factor and a multiple?
A: A factor divides into a number (it’s smaller or equal). A multiple is the result of multiplying a number (it’s larger or equal). Think of factors as building blocks and multiples as the results from a times table.
Q: In a word problem, how do I know whether to find the HCF or the LCM?
A: Look for keywords. HCF problems often involve splitting items into the largest or greatest number of equal groups. LCM problems often involve finding when repeating events will happen at the same time again, or the smallest number required.
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