能力傾向 - 數制

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數字

在十進位制數系中，有十個符號，即 0、1、2、3、4、5、6、7、8 和 9，稱為數字。一個數由這些數字組成的組表示，稱為數字。

面值

一個數字在數字中的面值是數字本身的值。例如，在 321 中，1 的面值是 1，2 的面值是 2，3 的面值是 3。

位值

一個數字在數字中的位值是數字乘以 10n 的值，其中 n 從 0 開始。例如，在 321 中

  • 1 的位值 = 1 x 100 = 1 x 1 = 1

  • 2 的位值 = 2 x 101 = 2 x 10 = 20

  • 3 的位值 = 3 x 102 = 3 x 100 = 300

    • 第 0 位數字稱為個位數字，是能力傾向測試中最常用的主題。

數字型別

  1. 自然數 - n > 0，其中 n 是計數數；[1,2,3...]

  2. 整數 - n ≥ 0，其中 n 是計數數；[0,1,2,3...]。

    3. 0 是唯一一個不是自然數的整數。

    每個自然數都是整數。

  3. 整數 - n ≥ 0 或 n ≤ 0，其中 n 是計數數；...,-3,-2,-1,0,1,2,3... 是整數。

    • 正整數 - n > 0；[1,2,3...]

    • 負整數 - n < 0；[-1,-2,-3...]

    • 非正整數 - n ≤ 0；[0,-1,-2,-3...]

    • 非負整數 - n ≥ 0；[0,1,2,3...]

    0 既不是正整數也不是負整數。

  4. 偶數 - n / 2 = 0，其中 n 是計數數；[0,2,4,...]

  5. 奇數 - n / 2 ≠ 0，其中 n 是計數數；[1,3,5,...]

  6. 質數 - 除了 1 之外只能被自身整除的數。

    7. 1 不是質數。

    要測試一個數 p 是否為質數，找到一個整數 k，使得 k > √p。獲取小於或等於 k 的所有質數，並將 p 除以這些質數中的每一個。如果沒有任何數能整除 p，則 p 是質數，否則它不是質數。

    Example: 191 is prime number or not?
Solution: 
Step 1 - 14 > √191
Step 2 - Prime numbers less than 14 are 2,3,5,7,11 and 13.
Step 3 - 191 is not divisible by any above prime number.
Result - 191 is a prime number.

Example: 187 is prime number or not?
Solution: 
Step 1 - 14 > √187
Step 2 - Prime numbers less than 14 are 2,3,5,7,11 and 13.
Step 3 - 187 is divisible by 11.
Result - 187 is not a prime number.

  7. 合數 - 大於 1 的非質數。例如，4、6、8、9 等。

    8. 1 既不是質數也不是合數。

    2 是唯一的偶質數。

  8. 互質數 - 如果兩個自然數的最大公約數為 1，則它們是互質數。例如，(2,3)、(4,5) 是互質數。

整除性

以下是檢查數字整除性的技巧。

  1. 能被 2 整除 - 如果一個數的個位數字是 0、2、4、6 或 8，則該數能被 2 整除。

    2. Example: 64578 is divisible by 2 or not?
Solution: 
Step 1 - Unit digit is 8.
Result - 64578 is divisible by 2.

Example: 64575 is divisible by 2 or not?
Solution: 
Step 1 - Unit digit is 5.
Result - 64575 is not divisible by 2.

  2. 能被 3 整除 - 如果一個數的各位數字之和能被 3 整除，則該數能被 3 整除。

    3. Example: 64578 is divisible by 3 or not?
Solution: 
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30 
which is divisible by 3.
Result - 64578 is divisible by 3.

Example: 64576 is divisible by 3 or not?
Solution: 
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28 
which is not divisible by 3.
Result - 64576 is not divisible by 3.

  3. 能被 4 整除 - 如果用最後兩位數字組成的數能被 4 整除，則該數能被 4 整除。

    4. Example: 64578 is divisible by 4 or not?
Solution: 
Step 1 - number formed using its last two digits is 78 
which is not divisible by 4.
Result - 64578 is not divisible by 4.

Example: 64580 is divisible by 4 or not?
Solution: 
Step 1 - number formed using its last two digits is 80 
which is divisible by 4.
Result - 64580 is divisible by 4.

  4. 能被 5 整除 - 如果一個數的個位數字是 0 或 5，則該數能被 5 整除。

    5. Example: 64578 is divisible by 5 or not?
Solution: 
Step 1 - Unit digit is 8.
Result - 64578 is not divisible by 5.

Example: 64575 is divisible by 5 or not?
Solution: 
Step 1 - Unit digit is 5.
Result - 64575 is divisible by 5.

  5. 能被 6 整除 - 如果一個數能被 2 和 3 整除，則該數能被 6 整除。

    6. Example: 64578 is divisible by 6 or not?
Solution: 
Step 1 - Unit digit is 8. Number is divisible by 2.
Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30 
which is divisible by 3.
Result - 64578 is divisible by 6.

Example: 64576 is divisible by 6 or not?
Solution: 
Step 1 - Unit digit is 8. Number is divisible by 2.
Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28 
which is not divisible by 3.
Result - 64576 is not divisible by 6.

  6. 能被 8 整除 - 如果用最後三位數字組成的數能被 8 整除，則該數能被 8 整除。

    7. Example: 64578 is divisible by 8 or not?
Solution: 
Step 1 - number formed using its last three digits is 578 
which is not divisible by 8.
Result - 64578 is not divisible by 8.

Example: 64576 is divisible by 8 or not?
Solution: 
Step 1 - number formed using its last three digits is 576 
which is divisible by 8.
Result - 64576 is divisible by 8.

  7. 能被 9 整除 - 如果一個數的各位數字之和能被 9 整除，則該數能被 9 整除。

    8. Example: 64579 is divisible by 9 or not?
Solution: 
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 9 = 31 
which is not divisible by 9.
Result - 64579 is not divisible by 9.

Example: 64575 is divisible by 9 or not?
Solution: 
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 5 = 27 
which is divisible by 9.
Result - 64575 is divisible by 9.

  8. 能被 10 整除 - 如果一個數的個位數字是 0，則該數能被 10 整除。

    9. Example: 64575 is divisible by 10 or not?
Solution: 
Step 1 - Unit digit is 5.
Result - 64578 is not divisible by 10.

Example: 64570 is divisible by 10 or not?
Solution: 
Step 1 - Unit digit is 0.
Result - 64570 is divisible by 10.

  9. 能被 11 整除 - 如果奇數位數字之和與偶數位數字之和的差為 0 或能被 11 整除，則該數能被 11 整除。

    10. Example: 64575 is divisible by 11 or not?
Solution: 
Step 1 - difference between sum of digits at odd places 
and sum of digits at even places = (6+5+5) - (4+7) = 5 
which is not divisible by 11.
Result - 64575 is not divisible by 11.

Example: 64075 is divisible by 11 or not?
Solution: 
Step 1 - difference between sum of digits at odd places 
and sum of digits at even places = (6+0+5) - (4+7) = 0.
Result - 64075 is divisible by 11.

除法技巧

  1. 如果一個數 n 能被兩個互質數 a、b 整除，則 n 能被 ab 整除。

  2. (a-b) 始終能整除 (an - bn)，如果 n 是自然數。

  3. (a+b) 始終能整除 (an - bn)，如果 n 是偶數。

  4. (a+b) 始終能整除 (an + bn)，如果 n 是奇數。

除法演算法

當一個數被另一個數除時

被除數 = (除數 x 商) + 餘數

級數

以下是基本數列的公式

  1. (1+2+3+...+n) = (1/2)n(n+1)

  2. (12+22+32+...+n2) = (1/6)n(n+1)(2n+1)

  3. (13+23+33+...+n3) = (1/4)n2(n+1)2

基本公式

這些是基本公式

(a + b)2 = a2 + b2 + 2ab

(a - b)2 = a2 + b2 - 2ab

(a + b)2 - (a - b)2 = 4ab

(a + b)2 + (a - b)2 = 2(a2 + b2)

(a2 - b2) = (a + b)(a - b)

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

(a3 + b3) = (a + b)(a2 -  ab + b2)

(a3 - b3) = (a - b)(a2 + ab + b2)

(a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
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