一.整数:integer,whole number

  1.因子:factor or divisor

  If x and y are integers and x≠0,x is a divisor (factor) of y provided that y=xn for some integer n. In this case y is also said to be divisible by x or to be a multiple of x. For example, 7 is a divisor or factor of 28 since 28=7•4, but 8 is not a divisor of 28 since there is no integer n such that 28=8n.Divisible adj.可以被整除的  multiple n.倍数

  2.商和余数:quotients and remainders

  余数和商都可以为0

  3.奇数和偶数:odd and even integers

  奇数和偶数都可以是负数;零一定是偶数

  4.质数和合数:prime numbers and composite numbers

  A prime number is a positive integer that has exactly two different positive divisors,1 and itself. For example, 2,3,5,7,11, and 13 are prime numbers, but 15 is not, since 15 has four different positive divisors, 1, 3, 5, and 15. The number 1 is not a prime number, since it has only one positive divisor. Every integer greater than 1 is either prime or can be uniquely expressed as a product of prime factors. For example, 14= (2) (7), 81= (3) (3) (3) (3), and 484= (2) (2) (11) (11).

  注:除了1和其本身外,还有其他因子的数叫合数。最小的质数为2,最小的合数为4,在讨论质数和合数时,都指正数。1和0既不是质数,也不是合数。

  5.整数中的重要概念:

  * Perfect square耆?椒绞??钊? = 32

  * Perfect cube 完全立方数,诸如8 = 23

  * the greatest common divisor 最大公约数

  几个数所公有的最大因子称最大公约数,诸如:48与36的公因子有1,2,3,4,6,12,其中12为最大公约数。

  * the least common multiple最小公倍数

  几个数所公有的最小倍数称最小公倍数,诸如:3,7和14的最小公倍数为42。

  *连续正整数的算术平均值也是首项和末项的算术平均值。

  同理,连续奇数与连续偶数的算术平均值也是首项和末项的算术平均值。

  * the properties of the number of factors因子个数的特性:

  1)当一个正整数n有奇数个因子,则n必为一完全平方数。

  2)除了n的平方根为其中一个因子外,小于n的平方根的因子与大于n的平方根的因子数相同。

  3)当某一正整数n有偶数因子时,则n必不是完全平方数,且大于n的平方根的因子与小于其的因子数相同。

  *因子数的求解公式:将整数n分解为质因子相乘的形式,然后将每个质因子的幂分别加1之后连乘所得的结果就是n的因子的个数。

  例:80的因子个数可以如下方式求得:80 = 2 4•5,则因子个数为(4+1)(1+1)= 10