Factors, multiples and primes

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Factors and Multiples

A factor of a number divides it exactly, leaving no remainder.
Every number has at least two factors: 1 and itself.
A multiple of a number is found by multiplying it by a whole number (1, 2, 3, ...).
Every number is a multiple of 1 and of itself.
The factors of a number are finite, but its multiples are infinite.
Example: the factors of 12 are 1, 2, 3, 4, 6, 12; the first multiples of 12 are 12, 24, 36, 48, ...

Prime Numbers

A prime number is a natural number greater than 1 with exactly two positive divisors: 1 and itself.
A natural number greater than 1 that is not prime is called a composite number.
1 is not prime because it has only one divisor (itself).
The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
2 is the only even prime; every other prime is odd.
Any prime greater than 5 ends in 1, 3, 7 or 9.
There are infinitely many primes.

Square and Cube Numbers

A square number is the result of multiplying a whole number by itself: n × n = n².
The first square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
A cube number is the result of multiplying a whole number by itself twice: n × n × n = n³.
The first cube numbers are 1, 8, 27, 64, 125, 216.
Square numbers have an odd number of factors; all other numbers have an even number of factors.

Prime Factorisation

Prime factorisation writes a number as a product of its prime factors.
The fundamental theorem of arithmetic says every integer greater than 1 is either prime or can be written as a unique product of primes (apart from order).
Use a factor tree: split the number into two factors, then split again until every branch ends in a prime.
Use repeated division: divide by the smallest prime that goes in exactly, and repeat until you reach 1.
Write the result using index notation, e.g. 360 = 2³ × 3² × 5.
Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Highest Common Factor (HCF)

The highest common factor (HCF) of two numbers is the largest number that divides both exactly.
Method 1: list all factors of each number and pick the largest shared one.
Method 2: write each number as a product of primes, then multiply the common prime factors (using the lowest power of each).
Example: 24 = 2³ × 3 and 36 = 2² × 3², so HCF = 2² × 3 = 12.
The HCF is never larger than the smaller of the two numbers.

Lowest Common Multiple (LCM)

The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
Method 1: list multiples of each number and pick the first shared one.
Method 2: write each number as a product of primes, then multiply the highest power of every prime that appears.
Example: 24 = 2³ × 3 and 36 = 2² × 3², so LCM = 2³ × 3² = 72.
The LCM is never smaller than the larger of the two numbers.

Venn Diagrams for HCF and LCM

Draw two overlapping circles, one for each number.
Put the shared prime factors in the overlap.
Put the remaining prime factors in the outer parts of each circle.
HCF = product of the primes in the overlap.
LCM = product of all the primes in the whole diagram.

Solving Word Problems with HCF and LCM

Use the HCF when splitting things into equal groups or finding the largest equal-sized piece.
Use the LCM when events repeat together or when finding when two things next coincide.
Look for key words: 'largest', 'greatest', 'maximum' often signal HCF; 'smallest', 'least', 'next together' often signal LCM.
Example (HCF): 24 pencils and 36 pens are shared equally into the greatest number of packs — 12 packs.
Example (LCM): one bus comes every 12 minutes and another every 18 minutes; they next leave together after 36 minutes.

Testing for Primality

Trial division checks whether a number n is divisible by any prime up to √(n).
If no prime up to √(n) divides n, then n is prime.
Example: to test 97, check primes up to √(97) ≈ 9.8: 2, 3, 5 and 7. None divide 97, so 97 is prime.
Primes are used in public-key cryptography, which relies on how hard it is to factor large numbers into primes.

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Câu hỏi luyện tập

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1.Which of these numbers is prime?
Easy
- A1
- B2
- C4
- D9
2.Which of these numbers is composite?
Easy
- A7
- B11
- C15
- D17
3.Which of these numbers is a square number?
Medium
- A8
- B9
- C12
- D18
4.Which of these numbers is a cube number?
Medium
- A6
- B8
- C16
- D24
5.Select all the prime numbers in this list. (select all that apply)
Medium
- A2
- B9
- C13
- D21
- E27
6.1 is a prime number.
Easy
True or false?
7.Every prime number greater than 2 is odd.
Easy
True or false?
8.Match each number to its correct description.
Easy
- 2
- 4
- 1
- 9
- the only even prime number
- a composite number
- neither prime nor composite
- a square number

Thêm câu hỏi

9.Put these numbers in order from smallest to largest.
Easy
- 6
- 2
- 15
- 3
10.Select all the factors of 24. (select all that apply)
Medium
- A1
- B2
- C3
- D4
- E5
- F6
11.Select all the multiples of 7 from this list. (select all that apply)
Medium
- A14
- B21
- C27
- D35
- E42
- F50
12.What is the highest common factor (HCF) of 24 and 36?
Medium
- A6
- B12
- C72
- D4
13.What is the lowest common multiple (LCM) of 4 and 6?
Medium
- A12
- B24
- C2
- D10
14.Every prime number greater than 5 ends in 1, 3, 7 or 9.
Easy
True or false?
15.Which of these numbers is a prime number?
Medium
- A51
- B57
- C53
- D55
16.Select all the numbers that are square numbers. (select all that apply)
Medium
- A1
- B4
- C9
- D16
- E20
- F25
17.Which of these numbers is a factor of 36?
Easy
- A8
- B9
- C11
- D14
18.Which of these numbers is a multiple of 12?
Easy
- A48
- B50
- C58
- D62
19.What is the highest common factor (HCF) of 16 and 24?
Medium
- A2
- B4
- C8
- D12
20.What is the lowest common multiple (LCM) of 5 and 7?
Medium
- A12
- B30
- C35
- D70
21.Select all the factors of 30. (select all that apply)
Medium
- A2
- B3
- C5
- D7
- E9
22.Select all the numbers that are multiples of 9. (select all that apply)
Medium
- A18
- B27
- C36
- D42
- E54
23.The number 2 is a prime number.
Easy
True or false?
24.The number 51 is a prime number.
Easy
True or false?
25.A bag contains 24 red beads and 36 blue beads. They are divided into identical groups with no beads left over, each group containing only one colour. What is the greatest possible number of beads in each group?
Medium
- A6
- B12
- C18
- D24
26.Two lighthouses flash at regular intervals. One flashes every 12 seconds and the other every 18 seconds. If they flash together at 8:00:00, at what time will they next flash together?
Medium
- A8:00:36
- B8:00:30
- C8:00:48
- D8:01:00
27.Which of the following statements are true? (select all that apply)
Medium
- AThe sum of two prime numbers is always even.
- BThe product of two prime numbers is always composite.
- CThe difference between two prime numbers is always odd.
- DEvery composite number has a prime factor less than or equal to its square root.
28.All prime numbers greater than 5 end in 1, 3, 7 or 9.
Easy
True or false?
29.Which of these numbers is the smallest prime number greater than 20?
Easy
- A21
- B23
- C25
- D27
30.Which of these numbers is a multiple of both 6 and 8?
Easy
- A12
- B18
- C24
- D36
31.A number is written as 2³ × 3 × 5². What is the value of this number?
Medium
- A150
- B300
- C600
- D900
32.Which of the following statements about the number 1 are true? (select all that apply)
Medium
- A1 is a prime number.
- B1 is a composite number.
- C1 has exactly one positive divisor.
- D1 is a factor of every natural number.
- E1 is a square number.
33.A prime number has exactly two positive divisors.
Easy
True or false?
34.Every natural number greater than 1 is either prime or composite.
Easy
True or false?
35.Match each number to its correct prime factorisation.
Medium
- 18
- 20
- 45
- 2 × 3²
- 2² × 5
- 3² × 5
36.The number 72 is written as a product of its prime factors. Which of these is correct?
Medium
- A2³ × 3²
- B2² × 3³
- C2³ × 3³
- D2² × 3²
37.Match each number to its prime factorisation written using index notation.
Medium
- 36
- 40
- 54
- 100
- 2² × 3²
- 2³ × 5
- 2 × 3³
- 2² × 5²
38.Match each number to its highest common factor (HCF) with 48.
Medium
- 18
- 32
- 45
- 50
- 6
- 16
- 3
- 2
39.Select all the numbers that are multiples of 8. (select all that apply)
Medium
- A48
- B56
- C84
- D96
- E108
40.Select all the numbers that have exactly two positive divisors. (select all that apply)
Medium
- A23
- B1
- C39
- D37
- E49
41.What is the highest common factor (HCF) of 12 and 18?
Easy
42.What is the lowest common multiple (LCM) of 6 and 8?
Medium
43.Complete the sentence about factors and multiples.
Easy
A ____ of a number divides it exactly, and a ____ of a number is found by multiplying it by an integer.
44.Write 60 as a product of its prime factors, using index notation if needed.
Medium
45.Complete the sentence about prime numbers.
Medium
A prime number is a natural number greater than ____ that has exactly ____ positive divisors.
46.What is the highest common factor (HCF) of 45 and 60?
Medium
47.What is the lowest common multiple (LCM) of 12 and 18?
Medium
48.Complete the definition of a prime number.
Easy
A prime number is a natural number greater than ____ that has exactly ____ positive divisors.
49.Complete the statement about prime factorisation.
Medium
The fundamental theorem of arithmetic says every natural number greater than 1 is either prime or can be factorised as a product of primes that is unique up to the ____ of the factors.
50.What name is given to a natural number greater than 1 that is not prime?
Medium
51.What is the smallest prime factor of 77?
Medium
52.What is the smallest prime number greater than 40?
Hard
53.Explain why 91 is not a prime number.
Medium
54.Explain why 51 is not a prime number.
Medium
55.Explain how to find the highest common factor (HCF) of 24 and 36.
Medium
56.Explain why 2 is the only even prime number.
Medium
57.Explain how to use trial division to test whether 97 is prime.
Hard
58.What is the highest common factor (HCF) of 12 and 18?
Easy
- A6
- B7
- C5
- D8
59.What is the lowest common multiple (LCM) of 6 and 8?
Medium
- A28
- B26
- C22
- D24
60.Write 60 as a product of its prime factors, using index notation if needed.
Medium
- A60
- B72
- C54
- D66
61.What is the highest common factor (HCF) of 45 and 60?
Medium
- A14
- B16
- C15
- D17
62.What is the lowest common multiple (LCM) of 12 and 18?
Medium
- A44
- B32
- C40
- D36
63.What is the smallest prime factor of 77?
Medium
- A8
- B9
- C7
- D6
64.What is the smallest prime number greater than 40?
Hard
- A41
- B37
- C49
- D45

Ghi chú ôn tập

## Factors, Multiples and Primes **Factors** of a number divide it exactly, leaving no remainder. Every number has 1 and itself as factors. Example: factors of 12 are 1, 2, 3, 4, 6, 12. **Multiples** are what you get when you multiply a number by positive integers. Example: multiples of 4 are 4, 8, 12, 16, ... **Prime numbers** are natural numbers greater than 1 that cannot be written as a product of two smaller natural numbers. They have exactly two positive divisors: 1 and themselves. 1 is not prime (it has only one divisor). Numbers greater than 1 that are not prime are **composite**. Example: 5 is prime; 4 = 2 × 2 and 6 = 2 × 3 are composite. The first 25 primes (all primes 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. Every prime except 2 is odd. All primes greater than 5 end in 1, 3, 7 or 9. **Square numbers** are the result of multiplying a number by itself: n × n = n^2. Example: 3^2 = 9. **Cube numbers** are the result of multiplying a number by itself twice: n × n × n = n^3. Example: 2^3 = 8. **Prime factorisation** means writing a number as a product of primes. The fundamental theorem of arithmetic says every natural number greater than 1 is either prime itself or can be factorised as a product of primes that is unique up to their order. Use **factor trees** or **repeated division** to find prime factors. Write repeated factors using index notation: 360 = 2^3 × 3^2 × 5. **Highest common factor (HCF)** is the largest factor shared by two or more numbers. **Lowest common multiple (LCM)** is the smallest multiple shared by two or more numbers. You can find both using **Venn diagrams**: place shared prime factors in the overlap, unique factors outside. HCF = product of the overlap; LCM = product of all the numbers in the Venn diagram. **Word problems**: use HCF when splitting into equal groups or finding the largest equal-sized piece; use LCM when events repeat together or when finding the smallest common amount. **Trial division** checks if a number n is prime by testing whether it is a multiple of any integer from 2 up to sqrt(n).

Factor: A natural number that divides another number exactly, leaving no remainder.
Multiple: The result of multiplying a number by a positive integer.
Prime number: A natural number greater than 1 that cannot be written as a product of two smaller natural numbers; it has exactly two positive divisors, 1 and itself.
Composite number: A natural number greater than 1 that is not prime; it can be written as a product of smaller natural numbers.
Square number: A number formed by multiplying an integer by itself, written as n^2.
Cube number: A number formed by multiplying an integer by itself twice, written as n^3.
Prime factorisation: Writing a number as a product of its prime factors, often using index notation.
Highest common factor (HCF): The largest factor that two or more numbers share.
Lowest common multiple (LCM): The smallest multiple that two or more numbers share.
Venn diagram: A diagram using overlapping circles to show common and unique prime factors when finding HCF and LCM.
Trial division: A method of checking primality by testing whether n is a multiple of any integer from 2 up to sqrt(n).
Fundamental theorem of arithmetic: Every natural number greater than 1 is either prime itself or can be factorised as a product of primes that is unique up to their order.

Thẻ ghi nhớ

What is a prime number?

A natural number greater than 1 that cannot be written as the product of two smaller natural numbers.

What is a composite number?

A natural number greater than 1 that is not prime; it can be written as a product of two smaller natural numbers.

Why is 1 not a prime number?

1 is not prime because it has only one positive divisor (itself), whereas a prime must have exactly two positive divisors: 1 and itself.

List the prime numbers less than 20.

2, 3, 5, 7, 11, 13, 17, 19.

What is the only even prime number?

2. Every even number greater than 2 is composite because it can be written as 2 × n/2.

What is a factor of a number?

A natural number that divides the number exactly, leaving no remainder.

What is a multiple of a number?

The result of multiplying that number by an integer. For example, multiples of 3 include 3, 6, 9, 12, …

What is a square number? Give an example.

A number that is the product of an integer with itself, e.g. 9 = 3 × 3.

What is a cube number? Give an example.

A number that is the product of an integer with itself twice, e.g. 8 = 2 × 2 × 2.

What is prime factorisation?

Writing a number as a product of its prime factors. For example, 12 = 2 × 2 × 3.

Write 360 as a product of its prime factors using index notation.

360 = 2³ × 3² × 5.

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

The largest number that divides exactly into both numbers.

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

The smallest number that is a multiple of both numbers.

How can a Venn diagram help find the HCF and LCM of two numbers?

Write the prime factors of each number in overlapping circles. The HCF is the product of the factors in the overlap; the LCM is the product of all the factors in the diagram.

How do you test whether a number n is prime using trial division?

Check whether n is divisible by any prime number up to √n. If none divide it exactly, n is prime.

What is the fundamental theorem of arithmetic?

Every natural number greater than 1 is either prime itself or can be written as a unique product of prime factors, apart from the order of the factors.

How many prime numbers are there?

There are infinitely many prime numbers. This was proved by Euclid around 300 BC.

What is Goldbach's conjecture?

The unsolved conjecture that every even integer greater than 2 can be expressed as the sum of two primes.

What is the twin prime conjecture?

The unsolved conjecture that there are infinitely many pairs of primes that differ by two.

Why are prime numbers important in cryptography?

Public-key cryptography relies on the difficulty of factoring large numbers into their prime factors.

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