Number patterns

# Number patterns ## - Dot representation of numbers (Triangular, Rectangular and square numbers ### Introduction Welcome to this exciting lesson on number patterns! Have you ever thought about how numbers can look like shapes? In this lesson, we will explore how we can represent numbers using dots to form different shapes like triangles, squares, and rectangles. This visual way of understanding numbers helps us discover amazing patterns and relationships. ### Key Concepts #### What are Number Patterns? A number pattern is a sequence of numbers that follows a certain rule. Understanding these rules helps us predict what numbers come next in the sequence. #### Triangular Numbers Triangular numbers are numbers that can be arranged to form an equilateral triangle when represented by dots. * **How they look:** * The 1st triangular number is 1 dot. (`•`) * The 2nd triangular number is 3 dots (1 + 2). (`•` on top, `• •` below) * The 3rd triangular number is 6 dots (1 + 2 + 3). (`•` on top, `• •` in middle, `• • •` below) * The 4th triangular number is 10 dots (1 + 2 + 3 + 4). * **Sequence:** 1, 3, 6, 10, 15, 21, ... * **Formula:** The \(n^{th}\) triangular number, denoted as \(T_n\), can be found using the formula: \[ T_n = \frac{n(n+1)}{2} \] Where \(n\) is the position of the number in the sequence (e.g., 1st, 2nd, 3rd, etc.). #### Square Numbers Square numbers are numbers that can be arranged to form a perfect square when represented by dots. * **How they look:** * The 1st square number is 1 dot. (`•`) * The 2nd square number is 4 dots (2 rows of 2). (`• •` on top, `• •` below) * The 3rd square number is 9 dots (3 rows of 3). * The 4th square number is 16 dots (4 rows of 4). * **Sequence:** 1, 4, 9, 16, 25, 36, ... * **Formula:** The \(n^{th}\) square number, denoted as \(S_n\), can be found using the formula: \[ S_n = n \times n = n^2 \] Where \(n\) is the position of the number in the sequence. #### Rectangular Numbers Rectangular numbers (specifically oblong numbers or pronic numbers in this context) are numbers that can be arranged to form a rectangle where the length and width differ by 1 (e.g., \(n\) rows and \(n+1\) columns of dots). * **How they look:** * The 1st rectangular number is 2 dots (1 row of 2). (`• •`) * The 2nd rectangular number is 6 dots (2 rows of 3). (`• • •` on top, `• • •` below) * The 3rd rectangular number is 12 dots (3 rows of 4). * The 4th rectangular number is 20 dots (4 rows of 5). * **Sequence:** 2, 6, 12, 20, 30, 42, ... * **Formula:** The \(n^{th}\) rectangular number, denoted as \(R_n\), can be found using the formula: \[ R_n = n \times (n+1) \] Where \(n\) is the position of the number in the sequence. * **Relationship to Triangular Numbers:** Notice that each rectangular number is twice the triangular number of the same position! For example, \(R_2 = 6\) and \(T_2 = 3\), so \(6 = 2 \times 3\). This is because \( n(n+1) = 2 \times \frac{n(n+1)}{2} \). ### Vocabulary * **Pattern:** A regular, repeated arrangement or sequence. * **Sequence:** A set of numbers arranged in a specific order following a rule. * **Triangular Number:** A number whose dots can form an equilateral triangle. * **Square Number:** A number whose dots can form a perfect square. * **Rectangular Number (Oblong/Pronic Number):** A number whose dots can form a rectangle with adjacent sides differing by one. ### Examples 1. **Finding the 5th Triangular Number:** Using the formula \( T_n = \frac{n(n+1)}{2} \): \( T_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = \frac{30}{2} = 15 \). Dot representation for 15 would be: 1 dot, 2 dots, 3 dots, 4 dots, 5 dots in increasing rows. 2. **Finding the 6th Square Number:** Using the formula \( S_n = n^2 \): \( S_6 = 6^2 = 6 \times 6 = 36 \). Dot representation for 36 would be a square of 6 rows and 6 columns. 3. **Finding the 4th Rectangular Number:** Using the formula \( R_n = n(n+1) \): \( R_4 = 4(4+1) = 4 \times 5 = 20 \). Dot representation for 20 would be a rectangle of 4 rows and 5 columns. ### Exercises 1. Draw the dot representation for the 5th triangular number. 2. What is the 7th square number? Show your calculation. 3. Is 45 a triangular number? Explain your answer. 4. Find the next two numbers in the sequence: 2, 6, 12, 20, ___, ___. 5. If a number is the 8th rectangular number, what is its value? How many dots would be in its rows and columns? ### Assignment 1. Calculate the 9th triangular, square, and rectangular numbers. 2. Can a number be both a square number and a triangular number (other than 1)? If yes, find the next number that is both. (Hint: Look at the sequences carefully!) 3. Explain in your own words how the formula for triangular numbers is related to the formula for rectangular numbers. 4. A classroom has 10 rows of desks, with each row having one more desk than the row before it (starting with 1 desk in the first row). How many desks are there in total? ### Summary Today, we learned about special number patterns that can be visualized using dots. We explored: * **Triangular Numbers:** Form triangles (1, 3, 6, 10, ...), calculated by \( T_n = \frac{n(n+1)}{2} \). * **Square Numbers:** Form squares (1, 4, 9, 16, ...), calculated by \( S_n = n^2 \). * **Rectangular Numbers (Oblong):** Form rectangles with sides differing by one (2, 6, 12, 20, ...), calculated by \( R_n = n(n+1) \). Understanding these patterns helps us see the beauty and order in mathematics. Keep practicing, and you'll become an expert in recognizing and working with these amazing numbers! ## Factors and multiples, Prime and composite numbers ### Introduction Welcome to this lesson on number patterns! In mathematics, understanding numbers goes beyond just counting. We often look for relationships between numbers. Today, we will explore two fundamental concepts: **factors** and **multiples**, which are essential building blocks for understanding many other mathematical ideas. We will also learn about special types of numbers called **prime numbers** and **composite numbers**, which are classified based on their factors. These concepts help us break down numbers and see their unique properties. ### Key Concepts #### What are Factors? Factors are numbers that divide another number exactly, leaving no remainder. Think of them as the numbers you multiply together to get a certain product. * **Definition:** A factor of a number \(n\) is a whole number that divides \(n\) with a remainder of zero. * **How to find factors:** To find the factors of a number, you can list all the pairs of numbers that multiply together to give that number. * **Properties of factors:** Every number has at least two factors: 1 and itself. #### What are Multiples? Multiples are the results you get when you multiply a number by other whole numbers (like 1, 2, 3, 4, and so on). * **Definition:** A multiple of a number is the product of that number and any whole number. * **How to find multiples:** To find the multiples of a number, you simply multiply it by 1, then by 2, then by 3, and so on. * **Properties of multiples:** A number has an infinite number of multiples. For example, multiples of 3 are \(3 \times 1 = 3\), \(3 \times 2 = 6\), \(3 \times 3 = 9\), and so on. #### What are Prime Numbers? Prime numbers are very special! They are the building blocks of all other whole numbers. * **Definition:** A prime number is a whole number greater than 1 that has **exactly two factors**: 1 and itself. * **Examples:** \(2, 3, 5, 7, 11, 13, 17, 19, \ldots\) * **Important note:** * The number 2 is the only even prime number. * The number 1 is NOT a prime number because it only has one factor (itself). #### What are Composite Numbers? Composite numbers are essentially the opposite of prime numbers. * **Definition:** A composite number is a whole number greater than 1 that has **more than two factors**. * **Examples:** \(4, 6, 8, 9, 10, 12, 14, 15, \ldots\) * **Important note:** The number 1 is neither prime nor composite. ### Vocabulary * **Factor:** A number that divides another number exactly without a remainder. * **Multiple:** The result of multiplying a number by another whole number. * **Prime Number:** A whole number greater than 1 with exactly two factors: 1 and itself. * **Composite Number:** A whole number greater than 1 with more than two factors. * **Divisor:** Another term for a factor. ### Examples #### Example 1: Finding Factors Find all the factors of 12. **Solution:** We need to find pairs of numbers that multiply to 12. \(1 \times 12 = 12\) \(2 \times 6 = 12\) \(3 \times 4 = 12\) The factors of 12 are 1, 2, 3, 4, 6, and 12. #### Example 2: Finding Multiples List the first 5 multiples of 4. **Solution:** We multiply 4 by the first five whole numbers: \(4 \times 1 = 4\) \(4 \times 2 = 8\) \(4 \times 3 = 12\) \(4 \times 4 = 16\) \(4 \times 5 = 20\) The first 5 multiples of 4 are 4, 8, 12, 16, and 20. #### Example 3: Identifying Prime and Composite Numbers Classify the numbers 7, 9, and 2 as prime or composite. **Solution:** * **For 7:** Its factors are 1 and 7 (exactly two factors). So, 7 is a **prime number**. * **For 9:** Its factors are 1, 3, and 9 (more than two factors). So, 9 is a **composite number**. * **For 2:** Its factors are 1 and 2 (exactly two factors). So, 2 is a **prime number**. ### Exercises 1. List all the factors of 15. 2. List the first 6 multiples of 5. 3. From the following list, identify all the prime numbers: 1, 2, 4, 5, 8, 11, 13, 16, 17. 4. From the following list, identify all the composite numbers: 3, 6, 7, 9, 10, 12, 19, 21. 5. Is the number 23 a prime or composite number? Explain your answer. ### Assignment 1. Find all the factors of 20. 2. Write down the first 7 multiples of 6. 3. Is 31 a prime or composite number? Show your working to explain. 4. List all prime numbers between 20 and 40. 5. List all composite numbers between 10 and 20. ### Summary In this lesson, we explored the fascinating world of number patterns through factors and multiples, and learned to classify numbers as prime or composite. We learned that: * **Factors** are numbers that divide another number exactly. * **Multiples** are products of a number with other whole numbers. * **Prime numbers** have exactly two factors: 1 and themselves (e.g., 2, 3, 5, 7). * **Composite numbers** have more than two factors (e.g., 4, 6, 8, 9). Understanding these basic concepts is crucial for future topics in mathematics, such as fractions, algebra, and number theory. Keep practicing to master identifying and working with these types of numbers! ## Prime factorisation of numbers # Prime factorisation of numbers\n\n## Introduction\nWelcome to this lesson on Prime Factorisation! In mathematics, numbers are like building blocks. Just as you can break down a building into its individual bricks, you can also break down numbers into their most basic components. This process is called factorisation, and when those components are special numbers called 'prime numbers', we call it **Prime Factorisation**. Understanding prime factorisation is a fundamental skill in mathematics that will help you with many other topics like finding the Lowest Common Multiple (LCM) and Highest Common Factor (HCF).\n\n## Key Concepts\n\n### Understanding Numbers\n\n* **Prime Numbers**\n * A prime number is a whole number greater than 1 that has exactly two positive divisors (factors): 1 and itself.\n * Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...\n * Note: \(2\) is the only even prime number.\n* **Composite Numbers**\n * A composite number is a whole number greater than 1 that has more than two positive divisors (factors).\n * Examples: 4 (factors: 1, 2, 4), 6 (factors: 1, 2, 3, 6), 8, 9, 10, ...\n * Note: The number \(1\) is neither prime nor composite.\n\n### What is Factorisation?\nFactorisation is the process of breaking down a number into a product of its factors. Factors are numbers that divide another number exactly, without leaving a remainder.\n\n* Example: For the number \(12\),\n * Factors are 1, 2, 3, 4, 6, 12.\n * We can write \(12 = 2 \times 6\) or \(12 = 3 \times 4\).\n\n### What is Prime Factorisation?\nPrime factorisation is the process of expressing a composite number as a product of its prime factors. This means that all the numbers multiplied together must be prime numbers.\n\n* Every composite number can be written as a unique product of prime numbers. This is a very important idea known as the **Fundamental Theorem of Arithmetic**.\n* Example: For the number \(12\),\n * \(12 = 2 \times 2 \times 3\)\n * In index form (using powers), this is written as \(2^2 \times 3\).\n\n### Methods of Prime Factorisation\nThere are two main methods to find the prime factorisation of a number:\n\n#### 1. Factor Tree Method\nIn this method, you break down the number into two factors, then break down those factors further until all the 'branches' end in prime numbers. The prime factors are the numbers at the end of the branches.\n\n#### 2. Division Method (Ladder Method)\nIn this method, you divide the given number by the smallest possible prime factor. Then, you divide the quotient by the smallest possible prime factor again, and continue this process until the quotient becomes 1. The prime factors are all the divisors you used.\n\n## Vocabulary\n\n* **Prime Number:** A natural number greater than 1 that has no positive divisors other than 1 and itself.\n* **Composite Number:** A natural number greater than 1 that has more than two positive divisors.\n* **Factor:** A number that divides another number exactly.\n* **Prime Factor:** A factor of a number that is also a prime number.\n* **Factorisation:** The process of breaking down a number into a product of its factors.\n* **Prime Factorisation:** The process of expressing a composite number as a product of its prime factors.\n\n## Examples\n\n**Example 1: Find the prime factorisation of 30.**\n\n* **Factor Tree Method:**\n ```\n 30\n / \\\n 2 15\n / \\\n 3 5\n ```\n So, \(30 = 2 \times 3 \times 5\).\n\n* **Division Method:**\n ```\n 2 | 30\n 3 | 15\n 5 | 5\n | 1\n ```\n So, \(30 = 2 \times 3 \times 5\).\n\n**Example 2: Find the prime factorisation of 48.**\n\n* **Factor Tree Method:**\n ```\n 48\n / \\\n 2 24\n / \\\n 2 12\n / \\\n 2 6\n / \\\n 2 3\n ```\n So, \(48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3\).\n\n* **Division Method:**\n ```\n 2 | 48\n 2 | 24\n 2 | 12\n 2 | 6\n 3 | 3\n | 1\n ```\n So, \(48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3\).\n\n**Example 3: Find the prime factorisation of 72 and express it in index form.**\n\n* **Using Division Method:**\n ```\n 2 | 72\n 2 | 36\n 2 | 18\n 3 | 9\n 3 | 3\n | 1\n ```\n So, \(72 = 2 \times 2 \times 2 \times 3 \times 3\).\n In index form, \(72 = 2^3 \times 3^2\).\n\n## Exercises\n\n1. List the first five prime numbers.\n2. Is 9 a prime or composite number? Explain why.\n3. Find the prime factors of 24 using the factor tree method.\n4. Express 50 as a product of its prime factors using the division method.\n5. Find the prime factorisation of 60 and write your answer in index form.\n\n## Assignment\n\n1. Find the prime factorisation of the following numbers using any method you prefer:\n a) 36\n b) 100\n c) 120\n d) 150\n e) 210\n2. Challenge: What is the smallest number that has 2, 3, and 5 as its prime factors?\n\n## Summary\n\nIn this lesson, you have learned that:\n* **Prime numbers** are numbers greater than 1 with only two factors: 1 and themselves (e.g., 2, 3, 5).\n* **Composite numbers** are numbers greater than 1 with more than two factors (e.g., 4, 6, 8).\n* **Prime factorisation** is the process of writing a composite number as a product of its prime factors.\n* Every composite number has a unique prime factorisation.\n* The two main methods for prime factorisation are the **Factor Tree Method** and the **Division Method**.\n* Prime factorisation is an important tool in understanding the building blocks of numbers. ## LCM and HCF (HCD) using prime factorisation ### Introduction Welcome to this lesson on finding the Lowest Common Multiple (LCM) and Highest Common Factor (HCF), also known as the Highest Common Divisor (HCD), using a powerful method called prime factorisation. This technique will help you work with numbers more efficiently, especially larger ones. By the end of this lesson, you will be able to confidently find the HCF and LCM of numbers using their prime factors. ### Key Concepts #### What are Prime Numbers? A prime number is a whole number greater than 1 that has only two factors: 1 and itself. * **Examples of Prime Numbers:** 2, 3, 5, 7, 11, 13, 17, 19, ... * **Numbers that are NOT Prime:** * 1 (It only has one factor). * 4 (Its factors are 1, 2, and 4). * 6 (Its factors are 1, 2, 3, and 6). #### Prime Factorisation Prime factorisation is the process of breaking down a number into a product of its prime factors. This means you express a number as a multiplication of only prime numbers. There are two common methods to find prime factorisation: * **1. Factor Tree Method:** * Start with the number at the top. * Break it down into any two factors. * Continue breaking down non-prime factors until all the branches end in prime numbers. * **Example: Prime factorisation of 30** ``` 30 / \ 2 15 / \ 3 5 ``` So, \( 30 = 2 \times 3 \times 5 \) * **2. Repeated Division (Ladder Method):** * Divide the number by the smallest possible prime number. * Continue dividing the quotient by the smallest possible prime number until the quotient is 1. * **Example: Prime factorisation of 30** ``` 2 | 30 3 | 15 5 | 5 | 1 ``` So, \( 30 = 2 \times 3 \times 5 \) #### Highest Common Factor (HCF) / Highest Common Divisor (HCD) using Prime Factorisation The HCF (or HCD) of two or more numbers is the largest number that divides exactly into all of them. **Steps to find HCF using prime factorisation:** 1. Find the prime factorisation of each number. 2. Identify all prime factors that are *common* to all the numbers. 3. For each common prime factor, take the *lowest power* (smallest exponent) it appears with in any of the factorisations. 4. Multiply these common prime factors (with their lowest powers) together to get the HCF. #### Lowest Common Multiple (LCM) using Prime Factorisation The LCM of two or more numbers is the smallest positive number that is a multiple of all the numbers. **Steps to find LCM using prime factorisation:** 1. Find the prime factorisation of each number. 2. Identify *all* prime factors that appear in *any* of the factorisations (both common and non-common factors). 3. For each unique prime factor, take the *highest power* (largest exponent) it appears with in any of the factorisations. 4. Multiply these prime factors (with their highest powers) together to get the LCM. ### Vocabulary * **Factor:** A number that divides another number exactly, leaving no remainder. For example, factors of 12 are 1, 2, 3, 4, 6, 12. * **Multiple:** The result of multiplying a number by an integer. For example, multiples of 3 are 3, 6, 9, 12, ... * **Prime Number:** A whole number greater than 1 that has exactly two factors: 1 and itself. * **Composite Number:** A whole number greater than 1 that has more than two factors. * **Prime Factorisation:** Expressing a number as a product of its prime numbers. * **HCF (Highest Common Factor) / HCD (Highest Common Divisor):** The largest factor that two or more numbers share. * **LCM (Lowest Common Multiple):** The smallest positive multiple that two or more numbers share. ### Examples #### Example 1: Find the HCF and LCM of 30 and 42 1. **Prime Factorisation:** * \( 30 = 2 \times 3 \times 5 \) * \( 42 = 2 \times 3 \times 7 \) 2. **Finding HCF:** * Common prime factors are 2 and 3. * Lowest power of 2 is \( 2^1 \). * Lowest power of 3 is \( 3^1 \). * HCF = \( 2 \times 3 = 6 \) 3. **Finding LCM:** * All unique prime factors are 2, 3, 5, and 7. * Highest power of 2 is \( 2^1 \). * Highest power of 3 is \( 3^1 \). * Highest power of 5 is \( 5^1 \). * Highest power of 7 is \( 7^1 \). * LCM = \( 2 \times 3 \times 5 \times 7 = 210 \) #### Example 2: Find the HCF and LCM of 24 and 36 1. **Prime Factorisation:** * \( 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1 \) * \( 36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 \) 2. **Finding HCF:** * Common prime factors are 2 and 3. * Lowest power of 2 in both factorisations is \( 2^2 \). * Lowest power of 3 in both factorisations is \( 3^1 \). * HCF = \( 2^2 \times 3^1 = 4 \times 3 = 12 \) 3. **Finding LCM:** * All unique prime factors are 2 and 3. * Highest power of 2 in any factorisation is \( 2^3 \). * Highest power of 3 in any factorisation is \( 3^2 \). * LCM = \( 2^3 \times 3^2 = 8 \times 9 = 72 \) ### Exercises Work out the following questions to test your understanding. 1. Find the prime factorisation of 48 and 72. 2. Using the prime factorisations from question 1, find the HCF of 48 and 72. 3. Using the prime factorisations from question 1, find the LCM of 48 and 72. 4. Find the HCF and LCM of 15, 25, and 30. ### Assignment Complete these questions at home to further practice your skills. 1. Find the prime factorisation for each of the following numbers: * a) 54 * b) 90 * c) 105 2. For each pair of numbers below, find their HCF and LCM using prime factorisation: * a) 18 and 27 * b) 40 and 60 * c) 75 and 100 3. Three different bells ring at intervals of 10, 15, and 20 minutes respectively. If they all ring together at 8:00 AM, at what time will they next ring together? ### Summary Today, you learned how to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) using prime factorisation. Remember these key points: * **Prime Factorisation** breaks numbers down into their fundamental prime building blocks. * For **HCF**, you multiply the *common* prime factors, each raised to its *lowest power* found in any of the factorisations. * For **LCM**, you multiply *all unique* prime factors (common and non-common), each raised to its *highest power* found in any of the factorisations. This method is very useful, especially when dealing with larger numbers, and provides a systematic way to find HCF and LCM. ## Square roots and cube roots of numbers using prime factorisation. ### Introduction Welcome to this lesson! Today, we're going to learn about square roots and cube roots. These are important concepts in mathematics that help us understand numbers better. We will discover a special method called **prime factorisation** to find these roots easily, especially for larger numbers. By the end of this lesson, you'll be able to find the square root and cube root of many numbers! ### Key Concepts #### What are Prime Numbers? A **prime number** is a whole number greater than 1 that has only two factors: 1 and itself. * Examples: 2, 3, 5, 7, 11, 13... #### What is Prime Factorisation? **Prime factorisation** is the process of breaking down a number into a product of its prime factors. Think of it like finding the building blocks of a number, where all the building blocks are prime numbers. * To find prime factors, you can use a factor tree or repeated division. * Example: The prime factors of 12 are \( 2 \times 2 \times 3 \). #### Square Roots A **square root** of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \( 5 \times 5 = 25 \). * The symbol for square root is \( \sqrt{} \). * A **perfect square** is a number whose square root is a whole number (e.g., 4, 9, 16, 25). ##### Finding Square Roots using Prime Factorisation 1. **Prime Factorise:** Break down the given number into its prime factors. 2. **Group in Pairs:** Group identical prime factors into pairs. 3. **Multiply Factors:** Take one factor from each pair and multiply them together. The result is the square root. #### Cube Roots A **cube root** of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). * The symbol for cube root is \( \sqrt[3]{} \). * A **perfect cube** is a number whose cube root is a whole number (e.g., 8, 27, 64, 125). ##### Finding Cube Roots using Prime Factorisation 1. **Prime Factorise:** Break down the given number into its prime factors. 2. **Group in Triplets:** Group identical prime factors into triplets (sets of three). 3. **Multiply Factors:** Take one factor from each triplet and multiply them together. The result is the cube root. ### Vocabulary * **Prime Number:** A whole number greater than 1 that has only two factors: 1 and itself. * **Prime Factorisation:** The process of breaking down a number into a product of its prime factors. * **Square Root:** A number that, when multiplied by itself, gives the original number. * **Perfect Square:** A number that is the square of a whole number. * **Cube Root:** A number that, when multiplied by itself three times, gives the original number. * **Perfect Cube:** A number that is the cube of a whole number. ### Examples #### Example 1: Find the square root of 36. 1. **Prime Factorise 36:** \( 36 = 2 \times 18 \) \( 18 = 2 \times 9 \) \( 9 = 3 \times 3 \) So, \( 36 = 2 \times 2 \times 3 \times 3 \) 2. **Group in Pairs:** \( (2 \times 2) \times (3 \times 3) \) 3. **Multiply Factors:** Take one from each pair: \( 2 \times 3 = 6 \) Therefore, \( \sqrt{36} = 6 \). #### Example 2: Find the square root of 144. 1. **Prime Factorise 144:** \( 144 = 2 \times 72 = 2 \times 2 \times 36 = 2 \times 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 2 \times 9 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \) 2. **Group in Pairs:** \( (2 \times 2) \times (2 \times 2) \times (3 \times 3) \) 3. **Multiply Factors:** Take one from each pair: \( 2 \times 2 \times 3 = 12 \) Therefore, \( \sqrt{144} = 12 \). #### Example 3: Find the cube root of 27. 1. **Prime Factorise 27:** \( 27 = 3 \times 9 \) \( 9 = 3 \times 3 \) So, \( 27 = 3 \times 3 \times 3 \) 2. **Group in Triplets:** \( (3 \times 3 \times 3) \) 3. **Multiply Factors:** Take one from the triplet: \( 3 \) Therefore, \( \sqrt[3]{27} = 3 \). #### Example 4: Find the cube root of 216. 1. **Prime Factorise 216:** \( 216 = 2 \times 108 = 2 \times 2 \times 54 = 2 \times 2 \times 2 \times 27 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \) 2. **Group in Triplets:** \( (2 \times 2 \times 2) \times (3 \times 3 \times 3) \) 3. **Multiply Factors:** Take one from each triplet: \( 2 \times 3 = 6 \) Therefore, \( \sqrt[3]{216} = 6 \). ### Exercises Try these questions to test your understanding: 1. Find the square root of the following numbers using prime factorisation: a) 64 b) 100 c) 225 2. Find the cube root of the following numbers using prime factorisation: a) 8 b) 64 c) 125 ### Assignment Continue practicing at home. Solve the following: 1. Find the square root of: a) 400 b) 729 2. Find the cube root of: a) 1000 b) 343 3. A square garden has an area of 441 square metres. What is the length of one side of the garden? ### Summary Today, we learned how to find square roots and cube roots of numbers using prime factorisation. Remember these key steps: * **For Square Roots:** Prime factorise the number, group factors in **pairs**, and multiply one factor from each pair. * **For Cube Roots:** Prime factorise the number, group factors in **triplets**, and multiply one factor from each triplet. This method is a powerful tool for working with numbers! ## - Even and odd numbers - Test for divisibility ### Introduction Welcome to this exciting lesson on number patterns! Today, we will explore two fundamental concepts in mathematics: even and odd numbers, and tests for divisibility. Understanding these concepts will help you work with numbers more efficiently and solve problems faster. By the end of this lesson, you will be able to identify even and odd numbers and quickly determine if a number can be divided exactly by another without actually performing the division. ### Key Concepts #### Even Numbers An **even number** is any whole number that can be divided by 2 without leaving a remainder. In other words, an even number is a multiple of 2. * **Characteristics:** * When you divide an even number by 2, the remainder is 0. * All even numbers end with the digits 0, 2, 4, 6, or 8. * **Examples:** 2, 4, 6, 8, 10, 12, 50, 124, 1000. #### Odd Numbers An **odd number** is any whole number that, when divided by 2, leaves a remainder of 1. * **Characteristics:** * When you divide an odd number by 2, the remainder is 1. * All odd numbers end with the digits 1, 3, 5, 7, or 9. * **Examples:** 1, 3, 5, 7, 9, 11, 23, 75, 999. #### Divisibility **Divisibility** means that a number can be divided by another number exactly, without leaving any remainder. If a number \(a\) is divisible by a number \(b\), then \(b\) is a factor of \(a\). #### Tests for Divisibility These tests are shortcuts to determine if a number is divisible by another number without doing long division. ##### Divisibility by 2 A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8). * **Example:** 346 is divisible by 2 because it ends in 6. ##### Divisibility by 3 A number is divisible by 3 if the sum of its digits is divisible by 3. * **Example:** Is 123 divisible by 3? Sum of digits: \(1+2+3=6\). Since 6 is divisible by 3, then 123 is divisible by 3. ##### Divisibility by 4 A number is divisible by 4 if the number formed by its last two digits is divisible by 4. * **Example:** Is 716 divisible by 4? The last two digits form the number 16. Since 16 is divisible by 4 (\(16 \div 4 = 4\)), then 716 is divisible by 4. ##### Divisibility by 5 A number is divisible by 5 if its last digit is 0 or 5. * **Example:** 50 and 135 are both divisible by 5. ##### Divisibility by 6 A number is divisible by 6 if it is divisible by both 2 and 3. * **Example:** Is 426 divisible by 6? * It ends in 6 (even), so it's divisible by 2. * Sum of digits: \(4+2+6=12\). Since 12 is divisible by 3, then 426 is divisible by 3. * Since it's divisible by both 2 and 3, it is divisible by 6. ##### Divisibility by 9 A number is divisible by 9 if the sum of its digits is divisible by 9. * **Example:** Is 819 divisible by 9? Sum of digits: \(8+1+9=18\). Since 18 is divisible by 9, then 819 is divisible by 9. ##### Divisibility by 10 A number is divisible by 10 if its last digit is 0. * **Example:** 270 and 1000 are both divisible by 10. ### Vocabulary * **Even Number:** A whole number that is a multiple of 2. * **Odd Number:** A whole number that is not a multiple of 2 (leaves a remainder of 1 when divided by 2). * **Divisibility:** The ability of one number to be divided by another number exactly, without a remainder. * **Remainder:** The amount left over after a division when one number is not perfectly divisible by another. * **Digit:** Any of the ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) used to write numbers. * **Sum of Digits:** The total you get when you add all the digits of a number together. ### Examples 1. **Identify Even/Odd:** Classify the following numbers as even or odd: 47, 92, 115, 200. * **Solution:** * 47: Ends in 7, so it's **odd**. * 92: Ends in 2, so it's **even**. * 115: Ends in 5, so it's **odd**. * 200: Ends in 0, so it's **even**. 2. **Test for Divisibility by 3:** Is 549 divisible by 3? * **Solution:** Sum of digits: \(5+4+9=18\). Since 18 is divisible by 3 (\(18 \div 3 = 6\)), then 549 is divisible by 3. 3. **Test for Divisibility by 4:** Is 1,028 divisible by 4? * **Solution:** Look at the number formed by the last two digits: 28. Since 28 is divisible by 4 (\(28 \div 4 = 7\)), then 1,028 is divisible by 4. 4. **Test for Divisibility by 6:** Is 732 divisible by 6? * **Solution:** * Ends in 2 (even), so divisible by 2. (Yes) * Sum of digits: \(7+3+2=12\). Since 12 is divisible by 3, it is divisible by 3. (Yes) * Since it is divisible by both 2 and 3, it is divisible by 6. ### Exercises 1. Determine if the following numbers are even or odd: * a) 35 * b) 88 * c) 129 * d) 450 2. Use the divisibility test to check if 768 is divisible by: * a) 2 * b) 3 * c) 4 * d) 6 3. Which of these numbers are divisible by 5? * 10, 27, 45, 103, 220 4. Is 9,360 divisible by 10? Explain your answer. 5. Find the sum of the digits of 6,102. Is this number divisible by 3? Is it divisible by 9? ### Assignment 1. Write down five even numbers and five odd numbers greater than 100. 2. Without performing actual division, state whether the following numbers are divisible by 2, 3, 5, or 10. You can list multiple factors for each. * a) 345 * b) 1,230 * c) 789 * d) 504 * e) 990 3. A number has 3 in its units place. Can it be divisible by 2? Can it be divisible by 5? Explain. 4. Is 2,148 divisible by 4? Show your working. 5. A number is divisible by both 2 and 3. Is it necessarily divisible by 6? Provide an example. ### Summary In this lesson, we explored the fascinating world of number patterns. We learned that **even numbers** are exactly divisible by 2 (ending in 0, 2, 4, 6, or 8), while **odd numbers** leave a remainder of 1 when divided by 2 (ending in 1, 3, 5, 7, or 9). We also discovered useful **divisibility tests** that help us quickly determine if a number can be divided by 2, 3, 4, 5, 6, 9, or 10 without needing to perform long division. These shortcuts are powerful tools that will make your mathematical calculations much easier and faster.