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Chapter 5 Notes Number Theory and Fractions Prime and Composite numbers A prime number is a number greater than 1 that has exactly two factors, itself and 1. A composite number is a number greater than 1 that has more than two factors. The numbers 1 and 0 are neither prime or composite Examples Prime: 3, 5, 11, 19 Composite: 12 (factors 1,2,3,4,6,12) and 25 (factors 1,5,25) Prime Factorization Every composite number can be written as the product of prime factors and this is called prime factorization. Use a factor tree to find the prime factors. To make a factor tree Start with the composite number Choose any 2 factors of that number Continue factoring until all the branches show prime numbers. Arrange the prime factors in order from least to greatest. Example 36 ↙↘ 9 4 ↙↘ ↙↘ 3 3 2 22 ×32 2 How to find equivalent fractions Multiply or divide the numerator (top number in a fraction) and the denominator (bottom number of the fraction) by the same number Examples 5 = 15 6 18 The denominator 6 is multiplied by 3 to get the new denominator 18. To find the equivalent numerator multiply the numerator 5 by 3 to get 15. 8 =1 32 4 The numerator 8 is divided by 8 to get the new numerator 1. To find the equivalent denominator divide the denominator 32 by 8 to get 4. To find the Greatest Common Factor (GCF) List all the factors of each number Find the common or shared factors Choose the greatest common factor Example 8: 1, 2, 4, 8 12: 1, 2, 3, 4, 6, 12 20: 1, 2, 4, 5, 10, 20 Common factors: 1, 2, 4 Greatest common factor: 4 To rename a fraction in simplest form: Find the GCF of the numerator and denominator Divide the numerator and denominator by their GCF If you divided and they are still not in simplest form, divide again. Example 20 →factors of 20: 1, 2, 4, 5, 10, 20 32 →factors of 32: 1, 2, 4, 8, 16, 32 GCF of 20 and 32 is 4. 20 ÷ 4 = 5 32 ÷ 4 8 Mixed Numbers and Improper Fractions A mixed number has a whole number and a fraction An improper fraction has a numerator equal to or greater than its denominator To rename a mixed number as an improper fraction: Multiply the whole number by the denominator Add the product to the numerator Write the sum over the denominator Example 3 ½ = (2 x 3) + 1 = 7 2 2 To rename an improper fraction as a whole number or as a mixed number Divide the numerator by the denominator If there is a remainder, write it over the denominator and express in simplest form Example 38 = 38÷4 = 9 R2 = 9 2/4 = 9 ½ 4 To find the least common multiple (LCM) Write out the multiples of both numbers starting with the larger one Go back and forth writing multiples until you find the lowest number, except 0, that is common to both Example Multiples of 12: 12, 24, 36, 48, 60 Multiples of 30: 30, 60 The LCM od 12 and 30 is 60. To compare fractions When fractions have a common denominator, compare their numerators to find the larger one. When the fractions have unlike denominators, find the two fractions least common denominator (least common multiple of denominators), rename them using this least common denominator, and compare the two new fractions. Example 5 > 4 6 5 The LCM of 5 and 6 is 30. 5/6 renamed with 30 as the denominator is 25/30. 4/5 renamed with 30 as the denominator is 24/30. This makes 5/6 greater than 4/5. To order fractions Finds the least common denominator (LCD) of all the fractions Rename each fraction using the LCD Compare the numerators and write the fractions in order If you are ordering mixed numbers, compare the whole numbers first Example 2 = 16 3 24 3 = 18 4 24 5 = 15 8 24 From least to greatest 5/8, 2/3, ¾ From greatest to least ¾, 2/3, 5/8 To rename a fraction as a decimal Divide the numerator by the denominator. Place the decimal point after the numerator and in the quotient Divide. Add zeros as needed. Example 0.75 4 3.00 To rename a mixed number as a decimal Separate the mixed number into a whole number part and a fraction part Rename the fraction part as a decimal Add the whole number part and the decimal Example 0.0625 16 1.0000 9 + 0.0625 = 9.0625 To rename a decimal as a fraction Read the given decimal The place value names the denominator The amount names the numerator. Write your fraction and put in simplest form Example 0.35 is thirty-five hundredths = 35 = 7 100 20