能力倾向 - 数制

数字

在十进制数系中，有十个符号，即 0、1、2、3、4、5、6、7、8 和 9，称为数字。一个数由这些数字组成的组表示，称为数字。

面值

一个数字在数字中的面值是数字本身的值。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

一个数字在数字中的位值是数字乘以 10ⁿ 的值，其中 n 从 0 开始。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 位数字称为个位数字，是能力倾向测试中最常用的主题。

数字类型

1. 自然数 - n > 0，其中 n 是计数数；[1,2,3...]

2. 整数 - n ≥ 0，其中 n 是计数数；[0,1,2,3...]。

0 是唯一一个不是自然数的整数。

每个自然数都是整数。

3. 整数 - n ≥ 0 或 n ≤ 0，其中 n 是计数数；...,-3,-2,-1,0,1,2,3... 是整数。

   • 正整数 - n > 0；[1,2,3...]

   • 负整数 - n < 0；[-1,-2,-3...]

   • 非正整数 - n ≤ 0；[0,-1,-2,-3...]

   • 非负整数 - n ≥ 0；[0,1,2,3...]

   0 既不是正整数也不是负整数。

4. 偶数 - n / 2 = 0，其中 n 是计数数；[0,2,4,...]

5. 奇数 - n / 2 ≠ 0，其中 n 是计数数；[1,3,5,...]

6. 质数 - 除了 1 之外只能被自身整除的数。

1 不是质数。

要测试一个数 p 是否为质数，找到一个整数 k，使得 k > √p。获取小于或等于 k 的所有质数，并将 p 除以这些质数中的每一个。如果没有任何数能整除 p，则 p 是质数，否则它不是质数。

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. 合数 - 大于 1 的非质数。例如，4、6、8、9 等。

1 既不是质数也不是合数。

2 是唯一的偶质数。

8. 互质数 - 如果两个自然数的最大公约数为 1，则它们是互质数。例如，(2,3)、(4,5) 是互质数。

整除性

以下是检查数字整除性的技巧。

1. 能被 2 整除 - 如果一个数的个位数字是 0、2、4、6 或 8，则该数能被 2 整除。

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 - 如果一个数的各位数字之和能被 3 整除，则该数能被 3 整除。

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 - 如果用最后两位数字组成的数能被 4 整除，则该数能被 4 整除。

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 - 如果一个数的个位数字是 0 或 5，则该数能被 5 整除。

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 - 如果一个数能被 2 和 3 整除，则该数能被 6 整除。

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 - 如果用最后三位数字组成的数能被 8 整除，则该数能被 8 整除。

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 - 如果一个数的各位数字之和能被 9 整除，则该数能被 9 整除。

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 - 如果一个数的个位数字是 0，则该数能被 10 整除。

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 - 如果奇数位数字之和与偶数位数字之和的差为 0 或能被 11 整除，则该数能被 11 整除。

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. 如果一个数 n 能被两个互质数 a、b 整除，则 n 能被 ab 整除。

2. (a-b) 始终能整除 (aⁿ - bⁿ)，如果 n 是自然数。

3. (a+b) 始终能整除 (aⁿ - bⁿ)，如果 n 是偶数。

4. (a+b) 始终能整除 (aⁿ + bⁿ)，如果 n 是奇数。

除法算法

当一个数被另一个数除时

级数

以下是基本数列的公式

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

这些是基本公式

(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)