ACT Math 2-整数的概念和性质

,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,2014/12/2,#,Arithmetic,算术,第二节 整数的概念和性质,.,The Concept of Integers (,整数的概念,),自然数,奇数,偶数,质数,合数,互质数,倍数和约数,公倍数,公约数或公因数,完全平方数,商和余数,连续整数,Natural Numbers,Odd Numbers,Even Numbers,Prime Numbers,Composite Numbers,Mutual Prime Numbers,Multiple and Divisor,Common Multiple,Common Divisor or Factor,Perfect Square,Quotients and Remainders,Consecutive Integers,.,The Concept of Integers (,整数的概念,),Natural Numbers (,自然数,):,Any of the numbers 0, 1, 2, 3, 4, that can be used to count the members of a set,Odd Numbers (,奇数,),Numbers that cannot be evenly divisible by 2,Even Numbers (,偶数,),Numbers that can be evenly divisible by 2,.,The Concept of Integers (,整数的概念,),Prime Numbers (,质数,),A natural number greater than 1 that has no positive divisors other than 1 and itself., such as 17.,Composite Numbers (,合数,),A natural number that can be factorized into two or more other positive integers,Mutual Prime Numbers (,互质数,),Two numbers with 1 only as their greatest common divisor,.,The Concept of Integers (,整数的概念,),Multiple and Divisor (,倍数和约数,),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 (,公倍数,),If a number is divisible by two or more integers, this number is a common multiple of these integers.,the least/lowest common multiple,最小公倍数,Common Divisor or Factor (,公约数或公因数,),If two or more integers are divisible by a number, this number is a common divisor of these integers.,the greatest common divisor,最大公约数,.,The Concept of Integers (,整数的概念,),Perfect Square (,完全平方数,),If the square root of an integer is still an integer, the original integer is a perfect square,Perfect cube,完全立方数,Quotients and Remainders (,商和余数,),The remainder is the integer left over after dividing one integer by another to produce an integer quotient.,Consecutive Integers (,连续整数,),.,The Properties of Integers (,整数的性质,),1. Positive and Negative (,正负性,),正整数,(positive integers),，也就是自然数,(natural numbers),如,1, 2, 3, ,负整数,(negative integers),如, -3, -2, -1,0,既不是整数也不是负数。,e.g.: Is x positive?,x,2, 1 = 0,X,3,+ 1 = 0,.,The Properties of Integers (,整数的性质,),2. Odd and Even (,奇偶性,),If,n,is an integer, then,is an even number and,is an odd number.,Odd odd = ?,Even even = ?,Odd even = ?,Even even even even = ?,Odd odd odd odd = ?,Odd odd odd odd even = ?,Even even = ?,.,The Properties of Integers (,整数的性质,),2. Odd and Even (,奇偶性,),任何一个大于,2,的偶数都可表示为两个质数的和。,2,个连续的自然数相乘必然为,2,的倍数，,3,个连续的自然数相乘必然为,6,的倍数，,若,3,个连续自然数的算术平均值为奇数，则这三个自然数的乘积必为,8,的倍数。,如：,(4+5+6)/3 = 5,，则,4 5 6 = 120,可被,8,整除。,ACT Practice,1. If x is an even integer, which of the following is an odd integer?,(A) 3x + 2 (B) 7x (C) 8x + 5,(D) x,2,(E) x,3,2. If a and b are integers and a b = 8, then (a + b) CANNOT be,(A) 0 (B) less than 6 (C) greater than 6,(C) an even integer (E) an odd integer,ACT Practice,3. If n is an integer greater than 2, which of the following CANNOT be an even integer?,(A) n,2,(B) n (n 1) (C) n 1,(D) n + 1 (E) 4n + 3,.,The Properties of Integers (,整数的性质,),3. Prime and Composite numbers (,质数和合数,),Note:,数字,1,既不是质数，也不是合数。,大于,2,的质数都是奇数，数字,2,是质数中唯一的偶数。,定理：任何一个大于,2,的偶数都可以表示为两个质数的和。,ACT Practice,1. Which of the following cannot be expressed as the sum of two prime numbers.,(A) 21 (B) 14 (C) 18 (D) 28 (E) 23,2. If the sum of two prime numbers is 40, what is the largest product of the two integers?,(A) 391 (B) 319 (C) 111,(D) 153 (E) 387,.,The Properties of Integers (,整数的性质,),4. Divisor and Multiple (,约数和倍数,),如果整数,a,能被整数,b,整除，则,a,能被,b,的约数整除。,0,为任何一个非零整数的倍数，,1,为任何一个整数的约数，任何一个质数有且只有,1,和它本身两个约数。,求最大公约数和最小公倍数,5. The Divisibility of Integers (,整数的整除特性,),Here are some shortcuts to determining divisibility by common numbers:,If the integer has this feature,Then it is divisible by,It ends in 0, 2, 4, 6 or 8,2,The sum of the digits is divisible by 3,3,The number formed by the last 2 digits is divisible by 4,4,The number end in 5 or 0,5,The number meets the tests for divisibility by 2 and 3,6,The number formed by the last 3 digits is divisible by 8,8,The sum of the digits is divisible by 9,9,The difference between the sum of odd digits and the sum of even digits is divisible by 11,11,6. The Properties of Consecutive Integers (,连续整数的性质,),Any two consecutive integers must be an odd integer and an even integer.,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.,The number of consecutive integers n, n+1, n+2, , n+k is k+1, and their sum is (n+k/2),(k+1),7. The Basic Properties of Square (,平方数的基本性质,),平方数的个位是,0, 1, 4, 5, 6, 9,之一；,偶平方数能被,4,整除；,奇平方数能被,8,整除余,1,，即它可写为,8k+1,，,k,为整数；,在相邻的两个自然数的平方之间不存在其他的完全平方数；,任何两个相邻自然数之积不是完全平方数；,两个奇数的平方之和不是完全平方数。,7.,自然数,n,次幂的尾数特征,尾数为,2,的数的幂的个位数一定以,2, 4, 8, 6,循环；,尾数为,3,的数的幂的个位数一定以,3, 9, 7, 1,循环；,尾数为,4,的数的幂的个位数一定以,4, 6,循环；,尾数为,6,的数的幂的个位数一定以,6,循环；,尾数为,7,的数的幂的个位数一定以,7, 9, 3, 1,循环；,尾数为,8,的数的幂的个位数一定以,8, 4, 2, 6,循环；,尾数为,9,的数的幂的个位数一定以,9, 1,循环；,Practice,：,1. If x and y are different prime numbers, each greater than 2, which of the following must be true?,x + y 91,x y is an even integer;,x/y is not an integer.,(A) II only (B) I and II only (C) I and III only,(D) II and III only (E) I, II, and III,Practice,：,2. Is x a whole number?,2x is even.,3x is odd.,3. How many even integers are between 17/4 and 47/2 ?,(A) 9 (B) 8 (C) 6 (D) 5 (E) 4,