Chap 1 GRE 和 GMAT 中的數學知識複習

Arithmetic

Integer (whole number)

• 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.

• quotients and remainders

  If x and y are positive integers, there exist unique integer q and r, called quotient and remainder respectively such that y = xq + r and 0 ≦ r < x.

  For example, when 28 is divided by 8, the quotient is 3 and the remainder is 4 since 28 = 8*3 + 4

  • Note that y is divisible by x if and only if the remainder r is 0.

    For example, 32 has a remainder of 0 when divided by 8 since 32 is divisible by 8.

  • Also note that when a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer.

    For example, 5 divided by 7 has the quotient 0 and the remainder 5 since 5 = 7*0 + 5.

• odd and even integers

  Any integer that is divisible by 2 is an even integer; the set of even integers is ..., -4, -2, 0, 2, 4, ....

  Integers that are not divisible by 2 are odd integers; ..., -3, -1, 1, 3, ... is the set of odd integers.

  • If at least one factor of a product of integers is even, the product is even;
  • If two integers are both even or both odd, their sum and difference are even.

    Otherwise, their sum and difference are odd.

  • 奇數和偶數都可以是負數，零一定是偶數
• 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.

    the smallest prime number is 2, the smallest composite number is 4

  • 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 = 3^4, and 484 = 2^2 * 11^2

  • 除了 1 和本身外，還有其他因子的數叫合數 (composite)
  • 在討論 prime 和 composite 時都指 positive integer，1 and 0 are not prime nor composite.
• important concepts
  • perfect square，完全平方數，ex: 9 = 3^2
  • prefect cube，完全立方數，ex: 8 = 2^3
  • the greatest common divisor (GCD)，最大公因數，ex: 48 和 36 的最大公因數是 12
  • the least common multiple (LCM)，最小公倍數，ex: 3, 7, 14 的最小公倍數是 42
  • 連續正整數的算數平均數也是首項和末項的算數平均數

    連續奇數與連續偶數的算數平均數也是首項和末項的算數平均數

  • 除了 √n 為其中一個因數外，小於 √n 的因數與大於 √n 的因數個數相同
  • 因數個數的求解公式： 將整數 n 分解為 質因數相乘的形式，然後將每個質因數的冪數分別加 1 相乘，所得結果即為 n 的因數個數

    ex: 80 = 2^4 * 5 => 因數個數為 (4+1) * (1+1) = 10

  • 整除特性
    • 被 2 整除： 個位定為偶數
    • 被 3 整除： 各位數的和能被 3 整除
    • 被 4 整除： 末兩位數能被 4 整除
    • 被 5 整除： 個位數為 0 or 5
    • 被 8 整除： 末三位數能被 8 整除
    • 被 9 整除： 各位數的和能被 9 整除
    • 被 11 整除： 奇數位和減去偶數位和之差值能被 11 整除
  • 整數 n 次冪方數之個位數特性 (其實僅需謹記，冪次方尾數會循環，其餘可自推)
    • 尾數為 2 的冪次方個位數: 以 2, 4, 8, 6 循環
    • 尾數為 3 的冪次方個位數: 以 3, 9, 7, 1 循環
    • 尾數為 4 的冪次方個位數: 以 4, 6 循環
    • 尾數為 7 的冪次方個位數: 以 7, 9, 3, 1 循環
    • 尾數為 8 的冪次方個位數: 以 8, 4, 2, 6 循環
    • 尾數為 9 的冪次方個位數: 以 9, 1 循環
    • 應用 ex: 7^123 和 3^321 的個位數哪個大

      123 / 4 = 30 .. 3 => 7^123 個位數為 3，321 / 4 = 80 .. 1 => 3^321 個位數為 3 => 一樣大

  • 大於 2 的偶數都能寫成兩個質數的和的形式

Fracrions

In a fraction n / d, n is the numerator and d is the denominator.

The denominator of a fraction can never be 0, because division by 0 is not defined.

Two fractions said to be equivalent if they represent the same number.

For example, 8/36 and 14/63 are equivalent since they both represent the number 2/9

In each case, the fraction is reduced to lowest terms by dividing both numerator adn denominator by their greatest common divisor (GCD)

• 倒數： reciprocal
• 帶分數： mixed number
• 假分數： improper fraction

Decimals

In the decimal system, the position of the period or decimal point determines the place value of the digits.

• ..., Thousands, Hundreds, Tens, (Ones or Units)： ...、千位、百位、十位、個位
• Tenths, Hundredths, Thousandths, ...： 十分位、百分位、千分位、...

Scientific Notation

Sometimes decimals are expressed as the product of a number with only one digit to the left of the decimal point and a power of 10. This is called scientific notation.

For example, 231 can be written as 2.31 * 10^2 and 0.231 can be written as 2.31 * 10^(-2)

Real numbers

• 虛數： imaginary number
• n is between 1 and 4 on the number line: 1 < n < 4

  n is between 1 and 4, inclusive: 1 ≦ n ≦ 4

• 絕對值： absolute value (某數在數軸上與零點之間的距離) ex: |-5| = |5| = 5
• If x, y, and z are real numbers, then
  • x + y = y + x and xy = yx
  • (x + y) + z = x + (y + z) and (xy)z = x(yz)
  • x(y + z) = xy + xz
  • If x and y are both positive, then x + y and xy are positive.
  • If x and y are both negative, then x + y is negative and xy is positive.
  • If x is positive and y is negative, then xy is negative.
  • If xy = 0, the x = 0 or y = 0
  • |x + y| ≦ |x| + |y|

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