ACT-Math-2-整數(shù)的概念和性質(zhì)PPT優(yōu)秀課件

1、Arithmetic算術(shù)算術(shù)第二節(jié)第二節(jié) 整數(shù)的概念和性質(zhì)整數(shù)的概念和性質(zhì)1. The Concept of Integers (整數(shù)的概念整數(shù)的概念) 自然數(shù) 奇數(shù) 偶數(shù) 質(zhì)數(shù) 合數(shù) 互質(zhì)數(shù) 倍數(shù)和約數(shù) 公倍數(shù) 公約數(shù)或公因數(shù) 完全平方數(shù) 商和余數(shù) 連續(xù)整數(shù) Natural Numbers Odd Numbers Even Numbers Prime Numbers Composite Numbers Mutual Prime Numbers Multiple and Divisor Common Multiple Common Divisor or Factor Perfect Squar

2、e Quotients and Remainders Consecutive Integers 2. The Concept of Integers (整數(shù)的概念整數(shù)的概念)Natural Numbers (自然數(shù)): Any of the numbers 0, 1, 2, 3, 4, that can be used to count the members of a set Odd Numbers (奇數(shù)) Numbers that cannot be evenly divisible by 2Even Numbers (偶數(shù)) Numbers that can be evenly div

3、isible by 23. The Concept of Integers (整數(shù)的概念整數(shù)的概念)Prime Numbers (質(zhì)數(shù)) A natural number greater than 1 that has no positive divisors other than 1 and itself., such as 17.Composite Numbers (合數(shù)) A natural number that can be factorized into two or more other positive integersMutual Prime Numbers (互質(zhì)數(shù)) Tw

4、o numbers with 1 only as their greatest common divisor4. The Concept of Integers (整數(shù)的概念整數(shù)的概念)Multiple and Divisor (倍數(shù)和約數(shù)) If one number can be exactly divided by a smaller number, the number is a multiple of a smaller number and the smaller number is a divisor of this number.Common Multiple (公倍數(shù)) If

5、 a number is divisible by two or more integers, this number is a common multiple of these integers. the least/lowest common multiple 最小公倍數(shù)Common Divisor or Factor (公約數(shù)或公因數(shù)) If two or more integers are divisible by a number, this number is a common divisor of these integers. the greatest common divis

6、or 最大公約數(shù)5. The Concept of Integers (整數(shù)的概念整數(shù)的概念)Perfect Square (完全平方數(shù)) If the square root of an integer is still an integer, the original integer is a perfect square Perfect cube 完全立方數(shù)Quotients and Remainders (商和余數(shù)) The remainder is the integer left over after dividing one integer by another to produ

7、ce an integer quotient.Consecutive Integers (連續(xù)整數(shù))6. The Properties of Integers (整數(shù)的性質(zhì)整數(shù)的性質(zhì))1. Positive and Negative (正負(fù)性) 正整數(shù) (positive integers)，也就是自然數(shù) (natural numbers), 如 1, 2, 3, 負(fù)整數(shù) (negative integers), 如, -3, -2, -10既不是整數(shù)也不是負(fù)數(shù)。e.g.: Is x positive?x2 1 = 0X3 + 1 = 07. The Properties of Integer

8、s (整數(shù)的性質(zhì)整數(shù)的性質(zhì))2. Odd and Even (奇偶性)(1) If n is an integer, then is an even number and is an odd number.(2)Odd odd = ?(3)Even even = ?(4)Odd even = ?(5) Even even even even = ?(6)Odd odd odd odd = ?(7) Odd odd odd odd even = ?(8)Even even = ?8. The Properties of Integers (整數(shù)的性質(zhì)整數(shù)的性質(zhì))2. Odd and Even (

9、奇偶性)(1) 任何一個(gè)大于2的偶數(shù)都可表示為兩個(gè)質(zhì)數(shù)的和。(2)2個(gè)連續(xù)的自然數(shù)相乘必然為2的倍數(shù)，3個(gè)連續(xù)的自然數(shù)相乘必然為6的倍數(shù)，(3)若3個(gè)連續(xù)自然數(shù)的算術(shù)平均值為奇數(shù)，則這三個(gè)自然數(shù)的乘積必為8的倍數(shù)。 如：(4+5+6)/3 = 5，則4 5 6 = 120可被8整除。9ACT Practice1. If x is an even integer, which of the following is an odd integer? (A) 3x + 2 (B) 7x (C) 8x + 5 (D) x2 (E) x32. If a and b are integers and a

10、b = 8, then (a + b) CANNOT be (A) 0 (B) less than 6 (C) greater than 6 (C) an even integer (E) an odd integer10ACT Practice3. If n is an integer greater than 2, which of the following CANNOT be an even integer? (A) n2 (B) n (n 1) (C) n 1 (D) n + 1 (E) 4n + 311. The Properties of Integers (整數(shù)的性質(zhì)整數(shù)的性質(zhì)

11、)3. Prime and Composite numbers (質(zhì)數(shù)和合數(shù))Note:(1) 數(shù)字1既不是質(zhì)數(shù)，也不是合數(shù)。(2)大于2的質(zhì)數(shù)都是奇數(shù)，數(shù)字2是質(zhì)數(shù)中唯一的偶數(shù)。定理：任何一個(gè)大于2的偶數(shù)都可以表示為兩個(gè)質(zhì)數(shù)的和。12ACT Practice1. Which of the following cannot be expressed as the sum of two prime numbers. (A) 21 (B) 14 (C) 18 (D) 28 (E) 232. If the sum of two prime numbers is 40, what is the lar

12、gest product of the two integers? (A) 391 (B) 319 (C) 111 (D) 153 (E) 38713. The Properties of Integers (整數(shù)的性質(zhì)整數(shù)的性質(zhì))4. Divisor and Multiple (約數(shù)和倍數(shù))(1) 如果整數(shù)a能被整數(shù)b整除，則a能被b的約數(shù)整除。(2)0為任何一個(gè)非零整數(shù)的倍數(shù)，1為任何一個(gè)整數(shù)的約數(shù)，任何一個(gè)質(zhì)數(shù)有且只有1和它本身兩個(gè)約數(shù)。求最大公約數(shù)和最小公倍數(shù)145. The Divisibility of Integers (整數(shù)的整除特性)Here are some shortcu

13、ts to determining divisibility by common numbers:If the integer has this featureThen it is divisible byIt ends in 0, 2, 4, 6 or 82The sum of the digits is divisible by 33The number formed by the last 2 digits is divisible by 44The number end in 5 or 05The number meets the tests for divisibility by 2

14、 and 36The number formed by the last 3 digits is divisible by 88The sum of the digits is divisible by 99The difference between the sum of odd digits and the sum of even digits is divisible by 1111156. The Properties of Consecutive Integers (連續(xù)整數(shù)的性質(zhì))(1) Any two consecutive integers must be an odd int

15、eger and an even integer.(2)One integer among any three consecutive integers can be divisible by three; thus, the product of these three consecutive integers can be divisible by 3.(3)The number of consecutive integers n, n+1, n+2, , n+k is k+1, and their sum is (n+k/2) (k+1)167. The Basic Properties

16、 of Square (平方數(shù)的基本性質(zhì))(1) 平方數(shù)的個(gè)位是0, 1, 4, 5, 6, 9之一；(2) 偶平方數(shù)能被4整除；(3) 奇平方數(shù)能被8整除余1，即它可寫(xiě)為8k+1，k為整數(shù)；(4) 在相鄰的兩個(gè)自然數(shù)的平方之間不存在其他的完全平方數(shù)；(5) 任何兩個(gè)相鄰自然數(shù)之積不是完全平方數(shù)；(6) 兩個(gè)奇數(shù)的平方之和不是完全平方數(shù)。177. 自然數(shù)n次冪的尾數(shù)特征(1) 尾數(shù)為2的數(shù)的冪的個(gè)位數(shù)一定以2, 4, 8, 6循環(huán)；(2) 尾數(shù)為3的數(shù)的冪的個(gè)位數(shù)一定以3, 9, 7, 1循環(huán)；(3) 尾數(shù)為4的數(shù)的冪的個(gè)位數(shù)一定以4, 6循環(huán)；(4) 尾數(shù)為6的數(shù)的冪的個(gè)位數(shù)一定以6循環(huán)；(5) 尾數(shù)為7的數(shù)的冪的個(gè)位數(shù)一定以7, 9, 3, 1循環(huán)；(6) 尾數(shù)為8的數(shù)的冪的個(gè)位數(shù)一定以8, 4, 2, 6循環(huán)；(7) 尾數(shù)為9的數(shù)的冪的個(gè)位數(shù)一定以9, 1循環(huán)；18Practice：1. If x and y are different prime numbers, each greater than 2, which of the following must be true?I.x + y 91II.x y is an even integer;III.x/