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Chapter 1 Factors and Multiples Numbers: Prime, composite, (integer, natural, whole, decimal, positive, negative, fractions, etc.) Prime factoris(z)ation Highest common factors (HCF) Least common multiple (LCM), why not Highest common multiple Square, square root, cube, cube root Class exercises Homework 1. Factors and multiples 18 = 1 x 18 = 2 x 9 = 3 x 6 = 6 x 3 = 9 x 2 = 18 x 1 1, 2, 3, 6, 9, 18 are all factors of 18 18 is a multiple of 1, 2, 3, 6, 9, 18 When 18 is divided by its factors 1, 2, 3, 6, 9, 18, the reminder is 0 divisible 2. Prime numbers and composite numbers Prime number: a number which has only two different factors, 1 and the number itself Examples: 2, 3, 5, 7… Composite number: a number has more than two different factors Examples: 4, 6, 8, 10, 15, … The number 1 is neither a prime number nor a composite number Goldbach’s Conjecture: Every even number greater than 2 can be expressed as the sum of two prime numbers, verify using the following numbers 16, 36, 64, 98 Two-digit prime numbers with reversed digits: 37 and 73, others? Twin prime numbers: prime numbers that differ by 2, like 5 and 7, list others? 3. Prime factorization Decomposition of a composite number into prime factors Example: 60 2 | 252 / \ 2 | 126 2 30 3| 63 / \ 3| 21 2 15 7 / \ 3 5 So, 60 = 2 x 2 x 3 x 5 252 = 2 X 2 X 3 X 3 X 7 = 2 2 X 32 X 7 4. Index Notation 5 x 5 x 5 is written 53 and is read 5 cubed or the cube of 5 5 x 5 x 5 x 5 is written as 54 and is read as 5 to the power of 4 a x a x a x a x … x a is written as an and is read as a to the power of n Exercises: Factorize 100, 125, 147, 216, 225, 360, 567, 648 5. Highest common factor (HCF) Greatest common factor (GCF) Case Study: How to cover a 30 cm by 36 cm sheet of paper completely with identical SQUARE patterns? Find the side of the largest possible square. Can we use 1 x 1 square? Or 2 x 2 square? The factors (prime factorization) of 30: 30 = 2 x 3 x 5 The factors (prime factorization) of 36: 36 = 22 x 32 So the HCF of 30 and 36 is 2 x 3 = 6, use 6 x 6 sheet to cover the 30 x 36 area Method 2: 2 | 30 36 3| 15 18 5 6 So the HCF is 2 x 3, there is no common factors between 5 and 6 bigger than 1 Exercises: Find HCF of the 60, 180, 210 6. Least common multiple (LCM) Case Study: Using a rectangular pattern measuring 9 cm by 12 cm to form a square, decide how many rectangular patterns needed The multiples of 9 are 9, 18, 27, 36…, The multiples of 12 are 12, 24, 36… The least common multiple of 9 and 12 is 36 Class exercises: find LCM of 30 and 36 30 = 2 x 3 x 5 36 = 22 x 32 Then the lease common multiple of 30 and 36 are 22 x 32 x 5 Method 2: 2 | 30 36, then the LCM is 2 x 3 x 5 x 6 3| 15 18 5 6 Exercise: Find LCM of 18, 24, 36 7. Squares and square roots, cubes and cube roots Find positive square root of 784 and 2025, find the cube roots of 512 and 5832 8. Workbook Summary and Practice questions: page 1- 3