Finding Multiples, HCF, and LCM

Questions and Answers

Explain how to find the first three multiples of the number 6.

The first three multiples of 6 are 6, 12, and 18, obtained by multiplying 6 by 1, 2, and 3.

What is the highest common factor (HCF) of 24 and 36, and how is it determined?

The HCF of 24 and 36 is 12, determined by listing their factors: 24 (1, 2, 3, 4, 6, 8, 12, 24) and 36 (1, 2, 3, 4, 6, 9, 12, 18, 36).

Describe the method to find the least common multiple (LCM) of 7 and 3.

To find the LCM of 7 and 3, list the multiples: multiples of 7 (7, 14, 21, 28...) and multiples of 3 (3, 6, 9, 12, 15, 18, 21...), and identify the smallest common multiple, which is 21.

Identify the factors of 15 and explain what makes them factors.

The factors of 15 are 1, 3, 5, and 15, as they can be multiplied in pairs to yield 15 and each divides 15 without any remainder. Signup and view all the answers

How can prime factorization be used to find the HCF of 30 and 42?

For 30, the prime factors are 2, 3, and 5; for 42, they are 2, 3, and 7. The HCF is found by multiplying the lowest powers of common primes: 2 and 3, which equals 6. Signup and view all the answers

Using the relationship with HCF, calculate the LCM of 8 and 12.

The HCF of 8 and 12 is 4, so the LCM is calculated as $\text{LCM}(8, 12) = \frac{8 \times 12}{4} = 24$. Signup and view all the answers

What method would you use to find the LCM of 10 and 15, and what is the answer?

Using the listing method, the multiples of 10 (10, 20, 30, 40...) and 15 (15, 30, 45...) show that the LCM is 30. Signup and view all the answers

Give an example of finding factors of the number 36 and explain its significance.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. These are significant because they represent all the integers that can combine to produce 36 through multiplication. Signup and view all the answers

Study Notes

Finding Multiples

Definition: Multiples of a number are obtained by multiplying the number by integers.
Method: For a number ( n ), its multiples can be found by calculating ( n \times 1, n \times 2, n \times 3, ) etc.
Example: Multiples of 3: 3, 6, 9, 12, 15...

Highest Common Factor (HCF)

Definition: The HCF of two or more numbers is the largest number that divides all of them without leaving a remainder.
Methods to Find HCF:
- Listing Factors: List all factors of the numbers and identify the greatest one.
- Prime Factorization: Break down each number into its prime factors and multiply the lowest powers of common primes.
- Division Method: Use the Euclidean algorithm; repeatedly divide and take remainders until reaching 0.
Example: HCF of 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- HCF = 6

Least Common Multiple (LCM)

Definition: The LCM of two or more numbers is the smallest number that is a multiple of all of them.
Methods to Find LCM:
- Listing Multiples: List multiples of each number and identify the smallest common one.
- Prime Factorization: Use prime factorization to identify the highest powers of all prime factors involved.
- Relationship with HCF: ( \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} )
Example: LCM of 4 and 5
- Multiples of 4: 4, 8, 12, 16, 20...
- Multiples of 5: 5, 10, 15, 20...
- LCM = 20

Definition of Factors

Definition: Factors of a number are integers that can be multiplied together to yield that number.
Characteristics:
- A factor must divide the number with no remainder.
- Every number has at least two factors: 1 and itself.
Example: Factors of 28: 1, 2, 4, 7, 14, 28.

Finding Multiples

Multiples are produced by multiplying a number by integers.
For any number ( n ), multiples can be calculated as ( n \times 1, n \times 2, n \times 3, ) etc.
Example for multiples of 3 includes: 3, 6, 9, 12, 15...

Highest Common Factor (HCF)

HCF refers to the largest number that can divide two or more numbers without leaving a remainder.
Methods to find HCF:
- Listing Factors: Identify all factors of each number, pinpoint the greatest.
- Prime Factorization: Factor each number into primes and multiply the lowest common powers.
- Division Method: Utilize the Euclidean algorithm through repeated division until finding a remainder of zero.
Example: For numbers 12 and 18:
- Factors of 12 are: 1, 2, 3, 4, 6, 12
- Factors of 18 are: 1, 2, 3, 6, 9, 18
- HCF of 12 and 18 is 6.

Least Common Multiple (LCM)

LCM represents the smallest number that is a multiple of two or more numbers.
Methods to find LCM:
- Listing Multiples: List multiples of all numbers, find the smallest one that appears in all lists.
- Prime Factorization: Identify maximum powers of all prime factors across the numbers.
- Relationship with HCF: Expressed as ( \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} ).
Example: For LCM of 4 and 5:
- Multiples of 4 include: 4, 8, 12, 16, 20...
- Multiples of 5 include: 5, 10, 15, 20...
- LCM of 4 and 5 is 20.

Definition of Factors

Factors are integers that multiply together to create a specific number.
A factor must evenly divide the number, leaving no remainder.
Every integer has at least two factors, which are 1 and the number itself.
Example: Factors of 28 are: 1, 2, 4, 7, 14, 28.

Description

This quiz covers the essential concepts of finding multiples, as well as determining the highest common factor (HCF) and least common multiple (LCM) of numbers. You will explore several methods for calculating HCF and LCM, along with practical examples to enhance your understanding. Perfect for students learning basic arithmetic concepts.