Factors, multiples, primes, HCF and LCM

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Factors and Multiples

A factor of a number is a whole number that divides it exactly, with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 exactly.
A multiple of a number is what you get when you multiply it by a whole number. For example, multiples of 3 are 3, 6, 9, 12, 15, ...
Every number has at least two factors: 1 and itself. For example, 7 has factors 1 and 7.
To find all factors of a number, test dividing by 1, 2, 3, ... up to the number itself. For example, factors of 12 are 1, 2, 3, 4, 6, 12.
Factors come in pairs that multiply to the number. For 12, the factor pairs are (1,12), (2,6), and (3,4).
Multiples go on forever. For example, multiples of 5 are 5, 10, 15, 20, ...
A number is a multiple of its factors. For example, 12 is a multiple of 3 and 4.

Prime and Composite Numbers

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. For example, 2, 3, 5, 7, 11 are prime.
A composite number is a whole number greater than 1 that has more than two factors. For example, 4, 6, 8, 9, 10 are composite.
The number 1 is not prime and not composite. It has only one factor, itself.
2 is the only even prime number. All other even numbers are composite because they are divisible by 2.
The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Every composite number can be broken down into prime factors. For example, 12 = 2 × 2 × 3.

Prime Factorization

Prime factorization is writing a number as a product of its prime factors. For example, 12 = 2 × 2 × 3.
You can use a factor tree to find prime factors. For example, for 18: 18 = 2 × 9, then 9 = 3 × 3, so 18 = 2 × 3 × 3.
The order of prime factors does not matter. For example, 12 = 2 × 2 × 3 = 3 × 2 × 2.
Every number greater than 1 has exactly one prime factorization (except for order). This is called the fundamental theorem of arithmetic.
To check if a number is prime, test dividing by primes up to its square root. For example, to check 29, test 2, 3, 5 (since √29 ≈ 5.4).
Prime factorization helps find HCF and LCM.

Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two numbers is the largest factor that divides both numbers exactly. It is also called the Greatest Common Divisor (GCD).
To find the HCF, list all factors of each number and pick the largest one they share. For example, factors of 12: 1,2,3,4,6,12; factors of 18: 1,2,3,6,9,18; HCF = 6.
You can also find HCF using prime factorization: multiply the common prime factors with the smallest powers. For 12 = 2² × 3 and 18 = 2 × 3², common primes are 2 and 3, so HCF = 2 × 3 = 6.
HCF is useful for simplifying fractions. For example, to simplify 12/18, divide top and bottom by HCF 6: 12/18 = 2/3.
If two numbers have no common prime factors, their HCF is 1. For example, HCF of 8 and 15 is 1.

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two numbers is the smallest multiple that both numbers share. It is also called the Least Common Multiple.
To find the LCM, list multiples of each number and pick the smallest one they share. For example, multiples of 4: 4,8,12,16,20,...; multiples of 6: 6,12,18,24,...; LCM = 12.
You can also find LCM using prime factorization: multiply all prime factors with the greatest powers. For 4 = 2² and 6 = 2 × 3, LCM = 2² × 3 = 12.
LCM is useful for finding a common denominator when adding or subtracting fractions. For example, to add 1/4 and 1/6, use LCM 12 as denominator.
If one number is a multiple of the other, the LCM is the larger number. For example, LCM of 3 and 12 is 12.

Square and Cube Numbers

A square number is the result of multiplying a whole number by itself. For example, 4 = 2 × 2, 9 = 3 × 3, 16 = 4 × 4.
Square numbers are written with a small 2: 2² = 4, 3² = 9, 4² = 16.
A cube number is the result of multiplying a whole number by itself three times. For example, 8 = 2 × 2 × 2, 27 = 3 × 3 × 3, 64 = 4 × 4 × 4.
Cube numbers are written with a small 3: 2³ = 8, 3³ = 27, 4³ = 64.
Square numbers can be shown as squares of dots, and cube numbers as cubes of dots.
The first few square numbers are 1, 4, 9, 16, 25, 36, ... The first few cube numbers are 1, 8, 27, 64, 125, ...

Using the Distributive Property

The distributive property says that a × (b + c) = a × b + a × c. For example, 4 × (9 + 2) = 4 × 9 + 4 × 2.
You can use it to rewrite a sum like 36 + 8 as a product. First, find the HCF of 36 and 8, which is 4.
Then write 36 + 8 as 4 × 9 + 4 × 2, and factor out the 4: 36 + 8 = 4 × (9 + 2).
This is called factoring out the common factor.
It can make calculations easier. For example, 36 + 8 = 44, and 4 × 11 = 44, so both give the same answer.

Prime Numbers Up to 100

The prime numbers less than 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
There are 25 prime numbers less than 100.
All prime numbers except 2 are odd.
All prime numbers greater than 5 end in 1, 3, 7, or 9. For example, 11, 13, 17, 19.
Numbers ending in 0, 2, 4, 6, or 8 are even and composite (except 2 itself).
Numbers ending in 0 or 5 are divisible by 5, so they are composite (except 5 itself).

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練習題

免費預覽——57 題中的 8 題。註冊即可查看全部。

1.Which of these numbers is a prime number?
Easy
- A1
- B2
- C4
- D9
2.The number 1 is a prime number.
Easy
True or false?
3.Which of these numbers is a multiple of 6?
Easy
- A18
- B20
- C22
- D26
4.Which of the following is a factor of 36?
Easy
- A7
- B8
- C9
- D10
5.Which of these numbers is a square number?
Medium
- A12
- B16
- C20
- D24
6.All even numbers greater than 2 are composite.
Easy
True or false?
7.Match each number to its correct type.
Medium
- 7
- 8
- 9
- prime
- composite
- square
8.Put these numbers in order from smallest to largest: 9, 12, 5, 8.
Medium
- 9
- 12
- 5
- 8

複習筆記

## Factors, Multiples, Primes, HCF and LCM – Quick Revision ### Key Ideas - **Factor**: a number that divides another number exactly (no remainder). Example: 3 is a factor of 12 because 12 ÷ 3 = 4. - **Multiple**: the result of multiplying a number by a whole number. Example: multiples of 5 are 5, 10, 15, 20, ... - **Prime number**: a number greater than 1 that has exactly two factors: 1 and itself. Example: 7 is prime because its only factors are 1 and 7. - **Composite number**: a number greater than 1 that has more than two factors. Example: 6 is composite because 1, 2, 3, and 6 are factors. - **1 is special**: it is neither prime nor composite because it has only one factor (itself). ### Finding Factors - List factor pairs: for 12, pairs are (1,12), (2,6), (3,4). - Check numbers up to the square root to avoid missing pairs. ### Prime Factorization - Write a number as a product of its prime factors. Use a factor tree. - Example: 18 = 2 × 3 × 3 = 2 × 3^2. ### HCF (Highest Common Factor) - The largest number that divides two or more numbers exactly. - Find by listing factors or using prime factorization. - Example: HCF of 12 and 18 is 6. - **Use**: simplifying fractions (e.g., 12/18 = 2/3). ### LCM (Lowest Common Multiple) - The smallest number that is a multiple of two or more numbers. - Find by listing multiples or using prime factorization. - Example: LCM of 4 and 6 is 12. - **Use**: finding a common denominator (e.g., 1/4 + 1/6 = 3/12 + 2/12 = 5/12). ### Distributive Property - Rewrite a sum like 36 + 8 as a product: 36 + 8 = 4 × (9 + 2). - This uses the HCF (4) as a common factor. ### Square and Cube Numbers - **Square number**: a number multiplied by itself. Example: 4^2 = 16. - **Cube number**: a number multiplied by itself three times. Example: 2^3 = 8. ### Quick Tips - Every prime number greater than 2 is odd. - All primes greater than 5 end in 1, 3, 7, or 9. - The first few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... ### Practice - Find all factors of 24. - Is 29 prime? Why? - Find HCF and LCM of 8 and 12. - Rewrite 45 + 15 using the distributive property.

Factor: A number that divides another number exactly, with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4.
Multiple: The result of multiplying a number by a whole number. For example, multiples of 5 are 5, 10, 15, 20, ...
Prime number: A number greater than 1 that has exactly two factors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7.
Composite number: A number greater than 1 that has more than two factors. For example, 6 is composite because it has factors 1, 2, 3, and 6.
HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. For example, the HCF of 12 and 18 is 6.
LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers. For example, the LCM of 4 and 6 is 12.
Square number: A number obtained by multiplying a number by itself. For example, 4^2 = 16, so 16 is a square number.
Cube number: A number obtained by multiplying a number by itself three times. For example, 2^3 = 8, so 8 is a cube number.

字卡

What is a factor of a number?

A factor of a number is a whole number that divides it exactly, with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4.

What is a multiple of a number?

A multiple of a number is the result of multiplying that number by a whole number. For example, multiples of 4 include 4, 8, 12, 16, and so on.

How are factors and multiples different?

Factors are numbers you multiply together to get another number. Multiples are what you get after multiplying a number by whole numbers. For example, 3 and 4 are factors of 12, and 12 is a multiple of 3 and 4.

What is a prime number?

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7.

What is a composite number?

A composite number is a whole number greater than 1 that has more than two factors. For example, 6 is composite because its factors are 1, 2, 3, and 6.

Why is 1 not a prime number?

1 is not prime because it has only one factor (itself), not two. Prime numbers must have exactly two different factors.

List all the factor pairs of 12.

The factor pairs of 12 are: 1 × 12, 2 × 6, and 3 × 4.

Write 36 as a product of its prime factors.

36 = 2 × 2 × 3 × 3, which can be written as 2² × 3².

What is the highest common factor (HCF) of two numbers?

The HCF is the largest number that divides both numbers exactly. For example, the HCF of 12 and 18 is 6.

Find the HCF of 12 and 18.

The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, 6, so the HCF is 6.

What is the lowest common multiple (LCM) of two numbers?

The LCM is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12.

Find the LCM of 4 and 6.

Multiples of 4: 4, 8, 12, 16, ... Multiples of 6: 6, 12, 18, ... The smallest common multiple is 12, so the LCM is 12.

How can you use the HCF to simplify a fraction?

Divide the top and bottom of the fraction by their HCF. For example, to simplify 12/18, divide both by 6 (the HCF) to get 2/3.

How can you use the LCM to add fractions with different denominators?

Use the LCM as the common denominator. For example, to add 1/4 and 1/6, the LCM is 12, so rewrite as 3/12 + 2/12 = 5/12.

Rewrite 36 + 8 using the distributive property.

36 + 8 = 4 × (9 + 2) because 4 × 9 = 36 and 4 × 2 = 8.

What is a square number?

A square number is the result of multiplying a whole number by itself. For example, 25 is a square number because 5 × 5 = 25.

What is a cube number?

A cube number is the result of multiplying a whole number by itself three times. For example, 8 is a cube number because 2 × 2 × 2 = 8.

Is 29 prime or composite? Explain.

29 is prime because its only factors are 1 and 29. It cannot be divided evenly by any other whole number.

What is the only even prime number?

2 is the only even prime number. Every other even number has 2 as a factor, so it is composite.

List the first 10 prime numbers.

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

What is the fundamental theorem of arithmetic?

It says that every whole number greater than 1 is either prime or can be written as a product of primes in exactly one way (ignoring order). For example, 30 = 2 × 3 × 5.

Find the HCF of 8 and 12 using prime factors.

8 = 2 × 2 × 2, 12 = 2 × 2 × 3. The common prime factors are 2 × 2, so the HCF is 4.

Find the LCM of 8 and 12 using prime factors.

8 = 2³, 12 = 2² × 3. Take the highest power of each prime: 2³ × 3 = 24. So the LCM is 24.

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