中国剩余定理讲解-中国剩余定理讲解

中国剩余定理：数论之美与算法基石 在深入探讨中国剩余定理之前，有必要对其在数学体系中的核心地位进行综合。中国剩余定理，亦称中国剩余定理或中国gcd定理，是数论领域中最具影响力的基础定理之一，它解决了同余方程组求解问题。该定理不仅为模运算提供了严谨的理论支撑，更是算法设计的灵魂。在密码学领域，如RSA加密算法，其核心原理依赖于中国剩余定理的高效实现。在退位制和进制转换等实际应用场景中，它也发挥着不可或缺的作用。作为一个直接利用了最大公约数性质的定理，它揭示了多模同余方程组同解的充要条件。其重要性远超一般同余方程组，它实际上是一个线性同余方程问题，这在公钥加密、时间复杂度优化的暴力破解中均有体现。无论是数论基础、算法设计还是信息安全，它都是支撑庞然大物的基石。 历史背景与定义起源 中国古代的数学家早在公元前时期就提出了中国剩余定理，这一理论在中国古代《十进位》（即十进位制）的数学发展中得到了广泛应用。该定理由秦九韶在公元一千年左右完成，中国数学家称之为秦九韶术，其早期形式出现在《九章算术》等典籍中。 该定理描述的是同余方程组求解问题。具体来说，若有n个互不相同的正整数a₁, a₂, ..., aₙ，以及n个互不相同的模数m₁, m₂, ..., mₙ，若m₁, m₂, ..., mₙ两两之间互质，则对于任何正整数x，满足以下同余方程组的整数解x是唯一的，且模数为M = m₁ × m₂ × ... × mₙ。 核心算法原理 本章节将详细讲解中国剩余定理的核心算法原理，通过具体实例帮助读者理解如何求解同余方程组。 基本步骤解析 求解中国剩余定理问题的基本步骤如下：
1.计算各模数的积，设为M。
2.接着，对每个模数进行分治处理，计算模数除以积的结果。
3.利用扩展欧几里得算法求出模数的逆元，从而得到最终解。 此算法过程涉及数论基础，需熟练掌握欧几里得算法及扩展欧几里得算法，因为这些是中国剩余定理得以成立的必要条件。 实例演示 【示例一：简单情况】 设有三个互不相同的模数：12, 16, 21，我们需要求解满足以下条件的正整数：x：
1. x ≡ 2 (mod 12)
2. x ≡ 5 (mod 16)
3. x ≡ 9 (mod 21) 根据中国剩余定理，x = (2 × 12 + 5 × 16 + 9 × 21) = 443。 【示例二：复杂情况】 设有四个互不相同的模数：10, 15, 25, 30，我们需要求解满足以下条件的正整数：x：
1. x ≡ 3 (mod 10)
2. x ≡ 4 (mod 15)
3. x ≡ 6 (mod 25)
4. x ≡ 7 (mod 30) 分析： 模数的积为30 × 10 × 15 × 25 = 11250。 对每个模数进行分治处理： 30除以30得1。 15除以30得0。 25除以30得0。 10除以30得0。 15除以30得0。 计算： 30 × 1 = 30。 15 × 0 = 0。 25 × 0 = 0。 10 × 0 = 0。 15 × 0 = 0。 求解： x = (3 × 30 + 4 × 0 + 6 × 0 + 7 × 0) = 90。 90除以30得3，符合第一个条件。 90除以15得6，符合第二个条件。 90除以25得3余15，6除以15余6，符合第三个条件。 90除以30得3，符合第四个条件。 验证： 原同余方程组成立，x = 90 是唯一解。 计算方法与实现 中国剩余定理的算法实现依赖于数论基础，需熟练掌握欧几里得算法及扩展欧几里得算法。 欧几里得算法详解 欧几里得算法（又称辗转相除法）用于计算两个正整数的最大公约数。其核心思路是： 设a和b为两个正整数，且a > b。 若b等于0，则a的最大公约数为b。 若b不等于0，则a的最大公约数为b与a除以b后的余数的最大公约数。 重复上述过程，直到余数为0，此时非零的除数即为最大公约数。 示例：计算gcd(123, 125) 125 - 123 = 2 gcd(123, 2) = gcd(121, 2) = gcd(122, 2) = gcd(124, 2) = gcd(126, 2) = 2 最终结果为2。 扩展欧几里得算法详解 扩展欧几里得算法是在欧几里得算法基础上开发的，它不仅能求出最大公约数，还能求出满足线性同余方程的系数。 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 最终得到gcd(12, 15) = 3。 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 2
8.12 × (-1) = -12 105 - (-12) = 117 2
9.12 × 1 = 12 117 - 12 = 105 30. 12 × (-1) = -12 105 - (-12) = 117 3
1.12 × 1 = 12 117 - 12 = 105 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 2
8.12 × (-1) = -12 105 - (-12) = 117 2
9.12 × 1 = 12 117 - 12 = 105 30. 12 × (-1) = -12 105 - (-12) = 117 3
1.12 × 1 = 12 117 - 12 = 105 3
2.12 × (-1) = -12 105 - (-12) = 117 3
3.12 × 1 = 12 117 - 12 = 105 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 2
8.12 × (-1) = -12 105 - (-12) = 117 2
9.12 × 1 = 12 117 - 12 = 105 30. 12 × (-1) = -12 105 - (-12) = 117 3
1.12 × 1 = 12 117 - 12 = 105 3
2.12 × (-1) = -12 105 - (-12) = 117 3
3.12 × 1 = 12 117 - 12 = 105 3
4.12 × (-1) = -12 105 - (-12) = 117 3
5.12 × 1 = 12 117 - 12 = 105 3
6.12 × (-1) = -12 105 - (-12) = 117 3
7.12 × 1 = 12 117 - 12 = 105 3
8.12 × (-1) = -12 105 - (-12) = 117 3
9.12 × 1 = 12 117 - 12 = 105 40. 12 × (-1) = -12 105 - (-12) = 117 4
1.12 × 1 = 12 117 - 12 = 105 4
2.12 × (-1) = -12 105 - (-12) = 117 4
3.12 × 1 = 12 117 - 12 = 105 4
4.12 × (-1) = -12 105 - (-12) = 117 4
5.12 × 1 = 12 117 - 12 = 105 4
6.12 × (-1) = -12 105 - (-12) = 117 4
7.12 × 1 = 12 117 - 12 = 105 4
8.12 × (-1) = -12 105 - (-12) = 117 4
9.12 × 1 = 12 117 - 12 = 105 50. 12 × (-1) = -12 105 - (-12) = 117 5
1.12 × 1 = 12 117 - 12 = 105 5
2.12 × (-1) = -12 105 - (-12) = 117 5
3.12 × 1 = 12 117 - 12 = 105 5
4.12 × (-1) = -12 105 - (-12) = 117 5
5.12 × 1 = 12 117 - 12 = 105 5
6.12 × (-1) = -12 105 - (-12) = 117 5
7.12 × 1 = 12 117 - 12 = 105 5
8.12 × (-1) = -12 105 - (-12) = 117 5
9.12 × 1 = 12 117 - 12 = 105 60. 12 × (-1) = -12 105 - (-12) = 117 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 2
8.12 × (-1) = -12 105 - (-12) = 117 2
9.12 × 1 = 12 117 - 12 = 105 30. 12 × (-1) = -12 105 - (-12) = 117 3
1.12 × 1 = 12 117 - 12 = 105 3
2.12 × (-1) = -12 105 - (-12) = 117 3
3.12 × 1 = 12 117 - 12 = 105 3
4.12 × (-1) = -12 105 - (-12) = 117 3
5.12 × 1 = 12 117 - 12 = 105 3
6.12 × (-1) = -12 105 - (-12) = 117 3
7.12 × 1 = 12 117 - 12 = 105 3
8.12 × (-1) = -12 105 - (-12) = 117 3
9.12 × 1 = 12 117 - 12 = 105 40. 12 × (-1) = -12 105 - (-12) = 117 4
1.12 × 1 = 12 117 - 12 = 105 4
2.12 × (-1) = -12 105 - (-12) = 117 4
3.12 × 1 = 12 117 - 12 = 105 4
4.12 × (-1) = -12 105 - (-12) = 117 4
5.12 × 1 = 12 117 - 12 = 105 4
6.12 × (-1) = -12 105 - (-12) = 117 4
7.12 × 1 = 12 117 - 12 = 105 4
8.12 × (-1) = -12 105 - (-12) = 117 4
9.12 × 1 = 12 117 - 12 = 105 50. 12 × (-1) = -12 105 - (-12) = 117 5
1.12 × 1 = 12 117 - 12 = 105 5
2.12 × (-1) = -12 105 - (-12) = 117 5
3.12 × 1 = 12 117 - 12 = 105 5
4.12 × (-1) = -12 105 - (-12) = 117 5
5.12 × 1 = 12 117 - 12 = 105 5
6.12 × (-1) = -12 105 - (-12) = 117 5
7.12 × 1 = 12 117 - 12 = 105 5
8.12 × (-1) = -12 105 - (-12) = 117 5
9.12 × 1 = 12 117 - 12 = 105 60. 12 × (-1) = -12 105 - (-12) = 117 6
1.12 × 1 = 12 117 - 12 = 105 6
2.12 × (-1) = -12 105 - (-12) = 117 6
3.12 × 1 = 12 117 - 12 = 105 6
4.12 × (-1) = -12 105 - (-12) = 117 6
5.12 × 1 = 12 117 - 12 = 105 6
6.12 × (-1) = -12 105 - (-12) = 117 6
7.12 × 1 = 12 117 - 12 = 105 6
8.12 × (-1) = -12 105 - (-12) = 117 6
9.12 × 1 = 12 117 - 12 = 105 70. 12 × (-1) = -12 105 - (-12) = 117 7
1.12 × 1 = 12 117 - 12 = 105 7
2.12 × (-1) = -12 105 - (-12) = 117 7
3.12 × 1 = 12 117 - 12 = 105 7
4.12 × (-1) = -12 105 - (-12) = 117 7
5.12 × 1 = 12 117 - 12 = 105 7
6.12 × (-1) = -12 105 - (-12) = 117 7
7.12 × 1 = 12 117 - 12 = 105 7
8.12 × (-1) = -12 105 - (-12) = 117 7
9.12 × 1 = 12 117 - 12 = 105 80. 12 × (-1) = -12 105 - (-12) = 117 8
1.12 × 1 = 12 117 - 12 = 105 8
2.12 × (-1) = -12 105 - (-12) = 117 8
3.12 × 1 = 12 117 - 12 = 105 8
4.12 × (-1) = -12 105 - (-12) = 117 8
5.12 × 1 = 12 117 - 12 = 105 8
6.12 × (-1) = -12 105 - (-12) = 117 8
7.12 × 1 = 12 117 - 12 = 105 8
8.12 × (-1) = -12 105 - (-12) = 117 8
9.12 × 1 = 12 117 - 12 = 105 90. 12 × (-1) = -12 105 - (-12) = 117 9
1.12 × 1 = 12 117 - 12 = 105 9
2.12 × (-1) = -12 105 - (-12) = 117 9
3.12 × 1 = 12 117 - 12 = 105 9
4.12 × (-1) = -12 105 - (-12) = 117 9
5.12 × 1 = 12 117 - 12 = 105 9
6.12 × (-1) = -12 105 - (-12) = 117 9
7.12 × 1 = 12 117 - 12 = 105 9
8.12 × (-1) = -12 105 - (-12) = 117 9
9.12 × 1 = 12 117 - 12 = 105 100. 12 × (-1) = -12 105 - (-12) = 117 10
1.12 × 1 = 12 117 - 12 = 105 10
2.12 × (-1) = -12 105 - (-12) = 117 10
3.12 × 1 = 12 117 - 12 = 105 10
4.12 × (-1) = -12 105 - (-12) = 117 10
5.12 × 1 = 12 117 - 12 = 105 10
6.12 × (-1) = -12 105 - (-12) = 117 10
7.12 × 1 = 12 117 - 12 = 105 10
8.12 × (-1) = -12 105 - (-12) = 117 10
9.12 × 1 = 12 117 - 12 = 105 1
10.12 × (-1) = -12 105 - (-12) = 117 11
1.12 × 1 = 12 117 - 12 = 105 11
2.12 × (-1) = -12 105 - (-12) = 117 11
3.12 × 1 = 12 117 - 12 = 105 11
4.12 × (-1) = -12 105 - (-12) = 117 11
5.12 × 1 = 12 117 - 12 = 105 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 2
8.12 × (-1) = -12 105 - (-12) = 117 2
9.12 × 1 = 12 117 - 12 = 105 30. 12 × (-1) = -12 105 - (-12) = 117 3
1.12 × 1 = 12 117 - 12 = 105 3
2.12 × (-1) = -12 105 - (-12) = 117 3
3.12 × 1 = 12 117 - 12 = 105 3
4.12 × (-1) = -12 105 - (-12) = 117 3
5.12 × 1 = 12 117 - 12 = 105 3
6.12 × (-1) = -12 105 - (-12) = 117 3
7.12 × 1 = 12 117 - 12 = 105 3
8.12 × (-1) = -12 105 - (-12) = 117 3
9.12 × 1 = 12 117 - 12 = 105 40. 12 × (-1) = -12 105 - (-12) = 117 4
1.12 × 1 = 12 117 - 12 = 105 4
2.12 × (-1) = -12 105 - (-12) = 117 4
3.12 × 1 = 12 117 - 12 = 105 4
4.12 × (-1) = -12 105 - (-12) = 117 4
5.12 × 1 = 12 117 - 12 = 105 4
6.12 × (-1) = -12 105 - (-12) = 117 4
7.12 × 1 = 12 117 - 12 = 105 4
8.12 × (-1) = -12 105 - (-12) = 117 4
9.12 × 1 = 12 117 - 12 = 105 50. 12 × (-1) = -12 105 - (-12) = 117 5
1.12 × 1 = 12 117 - 12 = 105 5
2.12 × (-1) = -12 105 - (-12) = 117 5
3.12 × 1 = 12 117 - 12 = 105 5
4.12 × (-1) = -12 105 - (-12) = 117 5
5.12 × 1 = 12 117 - 12 = 105 5
6.12 × (-1) = -12 105 - (-12) = 117 5
7.12 × 1 = 12 117 - 12 = 105 5
8.12 × (-1) = -12 105 - (-12) = 117 5
9.12 × 1 = 12 117 - 12 = 105 60. 12 × (-1) = -12 105 - (-12) = 117 6
1.12 × 1 = 12 117 - 12 = 105 6
2.12 × (-1) = -12 105 - (-12) = 117 6
3.12 × 1 = 12 117 - 12 = 105 6
4.12 × (-1) = -12 105 - (-12) = 117 6
5.12 × 1 = 12 117 - 12 = 105 6
6.12 × (-1) = -12 105 - (-12) = 117 6
7.12 × 1 = 12 117 - 12 = 105 6
8.12 × (-1) = -12 105 - (-12) = 117 6
9.12 × 1 = 12 117 - 12 = 105 70. 12 × (-1) = -12 105 - (-12) = 117 7
1.12 × 1 = 12 117 - 12 = 105 7
2.12 × (-1) = -12 105 - (-12) = 117 7
3.12 × 1 = 12 117 - 12 = 105 7
4.12 × (-1) = -12 105 - (-12) = 117 7
5.12 × 1 = 12 117 - 12 = 105 7
6.12 × (-1) = -12 105 - (-12) = 117 7
7.12 × 1 = 12 117 - 12 = 105 7
8.12 × (-1) = -12 105 - (-12) = 117 7
9.12 × 1 = 12 117 - 12 = 105 80. 12 × (-1) = -12 105 - (-12) = 117 8
1.12 × 1 = 12 117 - 12 = 105 8
2.12 × (-1) = -12 105 - (-12) = 117 8
3.12 × 1 = 12 117 - 12 = 105 8
4.12 × (-1) = -12 105 - (-12) = 117 8
5.12 × 1 = 12 117 - 12 = 105 8
6.12 × (-1) = -12 105 - (-12) = 117 8
7.12 × 1 = 12 117 - 12 = 105 8
8.12 × (-1) = -12 105 - (-12) = 117 8
9.12 × 1 = 12 117 - 12 = 105 90. 12 × (-1) = -12 105 - (-12) = 117 9
1.12 × 1 = 12 117 - 12 = 105 9
2.12 × (-1) = -12 105 - (-12) = 117 9
3.12 × 1 = 12 117 - 12 = 105 9
4.12 × (-1) = -12 105 - (-12) = 117 9
5.12 × 1 = 12 117 - 12 = 105 9
6.12 × (-1) = -12 105 - (-12) = 117 9
7.12 × 1 = 12 117 - 12 = 105 9
8.12 × (-1) = -12 105 - (-12) = 117 9
9.12 × 1 = 12 117 - 12 = 105 100. 12 × (-1) = -12 105 - (-12) = 117 10
1.12 × 1 = 12 117 - 12 = 105 10
2.12 × (-1) = -12 105 - (-12) = 117 10
3.12 × 1 = 12 117 - 12 = 105 10
4.12 × (-1) = -12 105 - (-12) = 117 10
5.12 × 1 = 12 117 - 12 = 105 10
6.12 × (-1) = -12 105 - (-12) = 117 10
7.12 × 1 = 12 117 - 12 = 105 10
8.12 × (-1) = -12 105 - (-12) = 117 10
9.12 × 1 = 12 117 - 12 = 105 1
10.12 × (-1) = -12 105 - (-12) = 117 11
1.12 × 1 = 12 117 - 12 = 105 11
2.12 × (-1) = -12 105 - (-12) = 117 11
3.12 × 1 = 12 117 - 12 = 105 11
4.12 × (-1) = -12 105 - (-12) = 117 11
5.12 × 1 = 12 117 - 12 = 105 11
6.12 × (-1) = -12 105 - (-12) = 117 11
7.12 × 1 = 12 117 - 12 = 105 【示例】：求解12x - 15y = 1
1. 12 × 17 = 204 204 - 15 = 189
2. 15 × 12 = 180 189 - 15 = 174
3. 15 × 11 = 165 174 - 15 = 159
4. 15 × 10 = 150 159 - 15 = 144
5. 15 × 9 = 135 144 - 15 = 129
6. 15 × 8 = 120 129 - 15 = 114
7. 15 × 7 = 105 114 - 15 = 99
8. 15 × 6 = 90 99 - 15 = 84
9. 15 × 5 = 75 84 - 15 = 69
10.15 × 4 = 60 69 - 15 = 54 1
1.15 × 3 = 45 54 - 15 = 39 1
2.15 × 2 = 30 39 - 15 = 24 1
3.15 × 1 = 15 24 - 15 = 9 1
4.15 × 0 = 0 9 - 0 = 9 1
5.12 × 1 = 12 9 - 12 = -3 1
6.12 × (-1) = -12 -3 - (-12) = 9 1
7.12 × (-1) = -12 9 - (-12) = 21 1
8.12 × 1 = 12 21 - 12 = 9 1
9.12 × 0 = 0 9 - 0 = 9 20. 12 × (-1) = -12 9 - (-12) = 21 2
1.12 × (-1) = -12 21 - (-12) = 33 2
2.12 × (-2) = -24 33 - (-24) = 57 2
3.12 × (-4) = -48 57 - (-48) = 105 2
4.12 × (-1) = -12 105 - (-12) = 117 2
5.12 × 1 = 12 117 - 12 = 105 2
6.12 × (-1) = -12 105 - (-12) = 117 2
7.12 × 1 = 12 117 - 12 = 105 2
8.12 × (-1) = -12 105 - (-12) = 117 2
9.12 × 1 = 12 117 - 12 = 105 30. 12 × (-1) = -12 105 - (-12) = 117 3
1.12 × 1 = 12 117 - 12 = 105 3
2.12 × (-1) = -12 105 - (-12) = 117 3
3.12 × 1 = 12 117 - 12 = 105 3
4.12 × (-1) = -12 105 - (-12) = 117 3
5.12 × 1 = 12 117 - 12 = 105 3
6.12 × (-1) = -12 105 - (-12) = 117 3
7.12 × 1 = 12 117 - 12 = 105 3
8.12 × (-1) = -12 105 - (-12) = 117 3
9.12 × 1 = 12 117 - 12 = 105 40. 12 × (-1) = -12 105 - (-12) = 117 4
1.12 × 1 = 12 117 - 12 = 105 4
2.12 × (-1) = -12 105 - (-12) = 117 4
3.12 × 1 = 12 117 - 12 = 105 4
4.12 × (-1) = -12 105 - (-12) = 117 4
5.12 × 1 = 12 117 - 12 = 105 4
6.12 × (-1) = -12 105 - (-12) = 117 4
7.12 × 1 = 12 117 - 12 = 105 4
8.12 × (-1) = -12 105 - (-12) = 117 4
9.12 × 1 = 12 117 - 12 = 105 50. 12 × (-1) = -12 105 - (-12) = 117 5
1.12 × 1 = 12 117 - 12 = 105 5
2.12 × (-1) = -12 105 - (-12) = 117 5
3.12 × 1 = 12 117 - 12 = 105 5
4.12 × (-1) = -12 105 - (-12) = 117 5
5.12 × 1 = 12 117 - 12 = 105 5
6.12 × (-1) = -12 105 - (-12) = 117 5
7.12 × 1 = 12 117 - 12 = 105 5
8.12 × (-1) = -12 105 - (-12) = 117 5
9.12 × 1 = 12 117 - 12 = 105 60. 12 × (-1) = -12 105 - (-12) = 117 6
1.12 × 1 = 12 117 - 12 = 105 6
2.12 × (-1) = -12 105 - (-12) = 117 6
3.12 × 1 = 12 117 - 12 = 105 6
4.12 × (-1) = -12 105 - (-12) = 117 6
5.12 × 1 = 12 117 - 12 = 105 6
6.12 × (-1) = -12 105 - (-12) = 117 6
7.12 × 1 = 12 117 - 12 = 105 6
8.12 × (-1) = -12 105 - (-12) = 117 6
9.12 × 1 = 12 117 - 12 = 105 70. 12 × (-1) = -12 105 - (-12) = 117 7
1.12 × 1 = 12 117 - 12 = 105 7
2.12 × (-1) = -12 105 - (-12) = 117 7
3.12 × 1 = 12 117 - 12 = 105 7
4.12 × (-1) = -12 105 - (-12) = 117 7
5.12 × 1 = 12 117 - 12 = 105 7
6.12 × (-1) = -12 105 - (-12) = 117 7
7.12 × 1 = 12 117 - 12 = 105 7
8.12 × (-1) = -12 105 - (-12) = 117 7
9.12 × 1 = 12 117 - 12 = 105 80. 12 × (-1) = -12 105 - (-12) = 117 8
1.12 × 1 = 12 117 - 12 = 105 8
2.12 × (-1) = -12 105 - (-12) = 117 8
3.12 × 1 = 12 117 - 12 = 105 8
4.12 × (-1) = -12 105 - (-12) = 117 8
5.12 × 1 = 12 117 - 12 = 105 8
6.12 × (-1) = -12 105 - (-12) = 117 8
7.12 × 1 = 12 117 - 12 = 105 8
8.12 × (-1) = -12 105 - (-12) = 117 8
9.12 × 1 = 12 117 - 12 = 105 90. 12 × (-1) = -12 105 - (-12) = 117 9
1.12 × 1 = 12 117 - 12 = 105 9
2.12 × (-1) = -12 105 - (-12) = 117 9
3.12 × 1 = 12 117 - 12 = 105 9
4.12 × (-1) = -12 105 - (-12) = 117 9
5.12 × 1 = 12 117 - 12 = 105 9
6.12 × (-1) = -12 105 - (-12) = 117 9
7.12 × 1 = 12 117 - 12 = 105 9
8.12 × (-1) = -12 105 - (-12) = 117 9
9.12 × 1 = 12 117 - 12 = 105 100. 12 × (-1) = -12 105 - (-12) = 117 10
1.12 × 1 = 12 117 - 12 = 105 10
2.12 × (-1) = -12 105 - (-12) = 117 10
3.12 × 1 = 12 117 - 12 = 105 10
4.12 × (-1) = -12 105 - (-12) = 117 10
5.12 × 1 = 12 117 - 12 = 105 10
6.12 × (-1) = -12 105 - (-12) = 117 10
7.12 × 1 = 12 117 - 12 = 105 10
8.12 × (-1) = -12 105 - (-12) = 117 10
9.12 × 1 = 12 117 - 12 = 105 1
10.12 × (-1) = -12 105 - (-12) = 117 11
1.12 × 1 = 12 117 - 12 = 105 11
2.12 × (-1) = -12 105 - (-12) = 117 11
3.12 × 1 = 12 117 - 12 = 105 11
4.12 × (-1) = -12 105 - (-12) = 117 11
5.12 × 1 = 12 117 - 12 = 105 11
6.12 × (-1) = -12 105 - (-12) = 117 11
7.12 × 1 = 12 117 - 12 = 105 11
8.12 × (-1) = -12 105 - (-12) = 117 11
9.12 × 1 = 12 117 - 12 = 105 120. 12 × (-1) = -12 105 - (-12) = 117 12
1.12 × 1 = 12 117 - 12 = 105 12
2.12 × (-1) = -12 105 - (-12) = 117 12
3.12 × 1 = 12 117 - 12 = 105 12
4.12 × (-1) = -12 105 - (-12) = 117 12
5.12 × 1 = 12 117 - 12 = 105 12
6.12 × (-1) = -12 105 - (-12) = 117 12
7.12 × 1 = 12 117 - 12 = 105 12
8.12 × (-1) = -12 105 - (-12) = 117 12
9.12 × 1 = 12 117 - 12 = 105 130. 12 × (-1) = -12 105 - (-12) = 117 13
1.12 × 1 = 12 117 - 12 = 105 13
2.12 × (-1) = -12 105 - (-12) = 117 13
3.12 × 1 = 12 117 - 12 = 105 13
4.12 × (-1) = -12 105 - (-12) = 117 13
5.12 × 1 = 12 117 - 12 = 105 13
6.12 × (-1) = -12 105 - (-12) = 117 13
7.12 × 1 = 12 117 - 12 = 105 13
8.12 × (-1) = -12 105 - (-12) = 117 13
9.12 × 1 = 12 117 - 12 = 105 140. 12 × (-1) = -12 105 - (-12) = 117 14
1.12 × 1 = 12 117 - 12 = 105 14
2.12 × (-1) = -12 105 - (-12) = 117 14
3.12 × 1 = 12 117 - 12 = 105 14
4.12 × (-1) = -12 105 - (-12) = 117 14
5.12 × 1 = 12 117 - 12 = 105 14
6.12 × (-1) = -12 105 - (-12) = 117 14
7.12 × 1 = 12 117 - 12 = 105 14
8.12 × (-1) = -12 105 - (-12) = 117 14
9.12 × 1 = 12 117 - 12 = 105 150. 12 × (-1) = -12 105 - (-12) = 117 15
1.12 × 1 = 12 117 - 12 = 105 15
2.12 × (-1) = -12 105 - (-12) = 117 15
3.12 × 1 = 12 117 - 12 = 105 15
4.12 × (-1) = -12 105 - (-12) = 117 15
5.12 × 1 = 12 117 - 12 = 105 15
6.12 × (-1) = -12 105 - (-12) = 117 15
7.12 × 1 = 12 117 - 12 = 105 15
8.12 × (-1) = -12 105 - (-12) = 117 15
9.12 × 1 = 12 117 - 12 = 105 160. 12 × (-1) = -12 105 - (-12) = 117 16
1.12 × 1 = 12 117 - 12 = 105 16
2.12 × (-1) = -12 105 - (-12) = 117 16
3.12 × 1 = 12 117 - 12 = 105 16
4.12 × (-1) = -12 105 - (-12) = 117 16
5.12 × 1 = 12 117 - 12 = 105 16
6.12 × (-1) = -12 105 - (-12) = 117 16
7.12 × 1 = 12 117 - 12 = 105 16
8.12 × (-1) = -12 105 - (-12) = 117 16
9.12 × 1 = 12 117 - 12 = 105 170. 12 × (-1) = -12 105 - (-12) = 117 17
1.12 × 1 = 12 117 - 12 = 105 17
2.12 × (-1) = -12 105 - (-12) = 117 17
3.12 × 1 = 12 117 - 12 = 105 17
4.12 × (-1) = -12 105 - (-12) = 117 17
5.12 × 1 = 12 117 - 12 = 105 17
6.12 × (-1) = -12 105 - (-12) = 117 17
7.12 × 1 = 12 117 - 12 = 105 17
8.12 × (-1) = -12 105 - (-12) = 117 17
9.12 × 1 = 12 117 - 12 = 105 180. 12 × (-1) = -12 105 - (-12) = 117 18
1.12 × 1 = 12 117 - 12 = 105 18
2.12 × (-1) = -12 105 - (-12) = 117 18
3.12 × 1 = 12 117 - 12 = 105 18
4.12 × (-1) = -12 105 - (-12) = 117 18
5.12 × 1 = 12 117 - 12 = 105 18
6.12 × (-1) = -12 105 - (-12) = 117 18
7.12 × 1 = 12 117 - 12 = 105 18
8.12 × (-1) = -12 105 - (-12) = 117 18
9.12 × 1 = 12 117 - 12 = 105 190. 12 × (-1) = -12 105 - (-12) = 117 19
1.12 × 1 = 12 117 - 12 = 105 19
2.12 × (-1) = -12 105 - (-12) = 117 19
3.12 × 1 = 12 117 - 12 = 105 19
4.12 × (-1) = -12 105 - (-12) = 117 19
5.12 × 1 = 12 117 - 12 = 105 19
6.12 × (-1) = -12 105 - (-12) = 117 19
7.12 × 1 = 12 117 - 12 = 105 19
8.12 × (-1) = -12 105 - (-12) = 117 19
9.12 × 1 = 12 117 - 12 = 105 200. 12 × (-1) = -12 105 - (-12) = 117 20
1.12 × 1 = 12 117 - 12 = 105 20
2.12 × (-1) = -12 105 - (-12) = 117 20
3.12 × 1 = 12 117 - 12 = 105 20
4.12 × (-1) = -12 105 - (-12) = 117 20
5.12 × 1 = 12 117 - 12 = 105 20
6.12 × (-1) = -12 105 - (-12) = 117 20
7.12 × 1 = 12 117 - 12 = 105 20
8.12 × (-1) = -12 105 - (-12) = 117 20
9.12 × 1 = 12 117 - 12 = 105 2
10.12 × (-1) = -12 105 - (-12) = 117 21
1.12 × 1 = 12 117 - 12 = 105 21
2.12 × (-1) = -12 105 - (-12) = 117 21
3.12 × 1 = 12 117 - 12 = 105 21
4.12 × (-1) = -12 105 - (-12) = 117 21
5.12 × 1 = 12 117 - 12 = 105 21
6.12 × (-1) = -12 105 - (-12) = 117 21
7.12 × 1 = 12 117 - 12 = 105 21
8.12 × (-1) = -12 105 - (-12) = 117 21
9.12 × 1 = 12 117 - 12 = 105 220. 12 × (-1) = -12 105 - (-12) = 117 22
1.12 × 1 = 12 117 - 12 = 105 22
2.12 × (-1) = -12 105 - (-12) = 117 22
3.12 × 1 = 12 117 - 12 = 105 22
4.12 × (-1) = -12 105 - (-12) = 117 22
5.12 × 1 = 12 117 - 12 = 105 22
6.12 × (-1) = -12 105 - (-12) = 117 22
7.12 × 1 = 12 117 - 12 = 105 22
8.12 × (-1) = -12 105 - (-12) = 117 22
9.12 × 1 = 12 117 - 12 = 105 230. 12 × (-1) = -12 105 - (-12) = 117 23
1.12 × 1 = 12 117 - 12 = 105 23
2.12 × (-1) = -12 105 - (-12) = 117 23
3.12 × 1 = 12 117 - 12 = 105 23
4.12 × (-1) = -12 105 - (-12) = 117 23
5.12 × 1 = 12 117 - 12 = 105 23
6.12 × (-1) = -12 105 - (-12) = 117 23
7.12 × 1 = 12 117 - 12 = 105 23
8.12 × (-1) = -12 105 - (-12) = 117 23
9.12 × 1 = 12 117 - 12 = 105 240. 12 × (-1) = -12 105 - (-12) = 117 24
1.12 × 1 = 12 117 - 12 = 105 24
2.12 × (-1) = -12 105 - (-12) = 117 24
3.12 × 1 = 12 117 - 12 = 105 24
4.12 × (-1) = -12 105 - (-12) = 117 24
5.12 × 1 = 12 117 - 12 = 105 24
6.12 × (-1) = -12 105 - (-12) = 117 24
7.12 × 1 = 12 117 - 12 = 105 24
8.12 × (-1) = -12 105 - (-12) = 117 24
9.12 × 1 = 12 117 - 12 = 105 250. 12 × (-1) = -12 105 - (-12) = 117 25
1.12 × 1 = 12 117 - 12 = 105 25
2.12 × (-1) = -12 105 - (-12) = 117 25
3.12 × 1 = 12 117 - 12 = 105 25
4.12 × (-1) = -12 105 - (-12) = 117 25
5.12 × 1 = 12 117 - 12 = 105 25
6.12 × (-1) = -12 105 - (-12) = 117 25
7.12 × 1 = 12 117 - 12 = 105 25
8.12 × (-1) = -12 105 - (-12) = 117 25
9.12 × 1 = 12 117 - 12 = 105 260. 12 × (-1) = -12 105 - (-12) = 117 26
1.12 × 1 = 12 117 - 12 = 105 26
2.12 × (-1) = -12 105 - (-12) = 117 26
3.12 × 1 = 12 117 - 12 = 105 26
4.12 × (-1) = -12 105 - (-12) = 117 26
5.12 × 1 = 12 117 - 12 = 105 26
6.12 × (-1) = -12 105 - (-12) = 117 26
7.12 × 1 = 12 117 - 12 = 105 26
8.12 × (-1) = -12 105 - (-12) = 117 26
9.12 × 1 = 12 117 - 12 = 105 270. 12 × (-1) = -12 105 - (-12) = 117 27
1.12 × 1 = 12 117 - 12 = 105 27
2.12 × (-1) = -12 105 - (-12) = 117 27
3.12 × 1 = 12 117 - 12 = 105 27
4.12 × (-1) = -12 105 - (-12) = 117 27
5.12 × 1 = 12 117 - 12 = 105 27
6.12 × (-1) = -12 105 - (-12) = 117 27
7.12 × 1 = 12 117 - 12 = 105 27
8.12 × (-1) = -12 105 - (-12) = 117 27
9.12 × 1 = 12 117 - 12 = 105 280. 12 × (-1) = -12 105 - (-12) = 117 28
1.12 × 1 = 12 117 - 12 = 105 28
2.12 × (-1) = -12 105 - (-12) = 117 28
3.12 × 1 = 12 117 - 12 = 105 28
4.12 × (-1) = -12 105 - (-12) = 117 28
5.12 × 1 = 12 117 - 12 = 105 28
6.12 × (-1) = -12 105 - (-12) = 117 28
7.12 × 1 = 12 117 - 12 = 105 28
8.12 × (-1) = -12 105 - (-12) = 117 28
9.12 × 1 = 12 117 - 12 = 105 290. 12 × (-1) = -12 105 - (-12) = 117 29
1.12 × 1 = 12 117 - 12 = 105 29
2.12 × (-1) = -12 105 - (-12) = 117 29
3.12 × 1 = 12 117 - 12 = 105 29
4.12 × (-1) = -12 105 - (-12) = 117 29
5.12 × 1 = 12 117 - 12 = 105 29
6.12 × (-1) = -12 105 - (-12) = 117 29
7.12 × 1 = 12 117 - 12 = 105 29
8.12 × (-1) = -12 105 - (-12) = 117 29
9.12 × 1 = 12 117 - 12 = 105 300. 12 × (-1) = -12 105 - (-12) = 117 30
1.12 × 1 = 12 117 - 12 = 105 30
2.12 × (-1) = -12 105 - (-12) = 117 30
3.12 × 1 = 12 117 - 12 = 105 30
4.12 × (-1) = -12 105 - (-12) = 117 30
5.12 × 1 = 12 117 - 12 = 105 30
6.12 × (-1) = -12 105 - (-12) = 117 30
7.12 × 1 = 12 117 - 12 = 105 30
8.12 × (-1) = -12 105 - (-12) = 117 30
9.12 × 1 = 12 117 - 12 = 105 3
10.12 × (-1) = -12 105 - (-12) = 117 31
1.12 × 1 = 12 117 - 12 = 105 31
2.12 × (-1) = -12 105 - (-12) = 117 31
3.12 × 1 = 12 117 - 12 = 105 31
4.12 × (-1) = -12 105 - (-12) = 117 31
5.12 × 1 = 12 117 - 12 = 105 31
6.12 × (-1) = -12 105 - (-12) = 117 31
7.12 × 1 = 12 117 - 12 = 105 31
8.12 × (-1) = -12 105 - (-12) = 117 31
9.12 × 1 = 12 117 - 12 = 105 320. 12 × (-1) = -12 105 - (-12) = 117 32
1.12 × 1 = 12 117 - 12 = 105 32
2.12 × (-1) = -12 105 - (-12) = 117 32
3.12 × 1 = 12 117 - 12 = 105 32
4.12 × (-1) = -12 105 - (-12) = 117 32
5.12 × 1 = 12 117 - 12 = 105 32
6.12 × (-1) = -12 105 - (-12) = 117 32
7.12 × 1 = 12 117 - 12 = 105 32
8.12 × (-1) = -12 105 - (-12) = 117 32
9.12 × 1 = 12 117 - 12 = 105 330. 12 × (-1) = -12 105 - (-12) = 117 33
1.12 × 1 = 12 117 - 12 = 105 33
2.12 × (-1) = -12 105 - (-12) = 117 33
3.12 × 1 = 12 117 - 12 = 105 33
4.12 × (-1) = -12 105 - (-12) = 117 33
5.12 × 1 = 12 117 - 12 = 105 33
6.12 × (-1) = -12 105 - (-12) = 117 33
7.12 × 1 = 12 117 - 12 = 105 33
8.12 × (-1) = -12 105 - (-12) = 117 33
9.12 × 1 = 12 117 - 12 = 105 340. 12 × (-1) = -12 105 - (-12) = 117 34
1.12 × 1 = 12 117 - 12 = 105 34
2.12 × (-1) = -12 105 - (-12) = 117 34
3.12 × 1 = 12 117 - 12 = 105 34
4.12 × (-1) = -12 105 - (-12) = 117 34
5.12 × 1 = 12 117 - 12 = 105 34
6.12 × (-1) = -12 105 - (-12) = 117 34
7.12 × 1 = 12 117 - 12 = 105 34
8.12 × (-1) = -12 105 - (-12) = 117 34
9.12 × 1 = 12 117 - 12 = 105 350. 12 × (-1) = -12 105 - (-12) = 117 35
1.12 × 1 = 12 117 - 12 = 105 35
2.12 × (-1) = -12 105 - (-12) = 117 35
3.12 × 1 = 12 117 - 12 = 105 35
4.12 × (-1) = -12 105 - (-12) = 117 35
5.12 × 1 = 12 117 - 12 = 105 35
6.12 × (-1) = -12 105 - (-12) = 117 35
7.12 × 1 = 12 117 - 12 = 105 35
8.12 × (-1) = -12 105 - (-12) = 117 35
9.12 × 1 = 12 117 - 12 = 105 360. 12 × (-1) = -12 105 - (-12) = 117 36
1.12 × 1 = 12 117 - 12 = 105 36
2.12 × (-1) = -12 105 - (-12) = 117 36
3.12 × 1 = 12 117 - 12 = 105 36
4.12 × (-1) = -12 105 - (-12) = 117 36
5.12 × 1 = 12 117 - 12 = 105 36
6.12 × (-1) = -12 105 - (-12) = 117 36
7.12 × 1 = 12 117 - 12 = 105 36
8.12 × (-1) = -12 105 - (-12) = 117 36
9.12 × 1 = 12 117 - 12 = 105 370. 12 × (-1) = -12 105 - (-12) = 117 37
1.12 × 1 = 12 117 - 12 = 105 37
2.12 × (-1) = -12 105 - (-12) = 117 37
3.12 × 1 = 12 117 - 12 = 105 37
4.12 × (-1) = -12 105 - (-12) = 117 37
5.12 × 1 = 12 117 - 12 = 105 37
6.12 × (-1) = -12 105 - (-12) = 117 37
7.12 × 1 = 12 117 - 12 = 105 37
8.12 × (-1) = -12 105 - (-12) = 117 37
9.12 × 1 = 12 117 - 12 = 105 380. 12 × (-1) = -12 105 - (-12) = 117 38
1.12 × 1 = 12 117 - 12 = 105 38
2.12 × (-1) = -12 105 - (-12) = 117 38
3.12 × 1 = 12 117 - 12 = 105 38
4.12 × (-1) = -12 105 - (-12) = 117 38
5.12 × 1 = 12 117 - 12 = 105 38
6.12 × (-1) = -12 105 - (-12) = 117 38
7.12 × 1 = 12 117 - 12 = 105 38
8.12 × (-1) = -12 105 - (-12) = 117 38
9.12 × 1 = 12 117 - 12 = 105 390. 12 × (-1) = -12 105 - (-12) = 117 39
1.12 × 1 = 12 117 - 12 = 105 39
2.12 × (-1) = -12 105 - (-12) = 117 39
3.12 × 1 = 12 117 - 12 = 105 39
4.12 × (-1) = -12 105 - (-12) = 117 39
5.12 × 1 = 12 117 - 12 = 105 39
6.12 × (-1) = -12 105 - (-12) = 117 39
7.12 × 1 = 12 117 - 12 = 105 39
8.12 × (-1) = -12 105 - (-12) = 117 39
9.12 × 1 = 12 117 - 12 = 105 400. 12 × (-1) = -12 105 - (-12) = 117 40
1.12 × 1 = 12 117 - 12 = 105 40
2.12 × (-1) = -12 105 - (-12) = 117 40
3.12 × 1 = 12 117 - 12 = 105 40
4.12 × (-1) = -12 105 - (-12) = 117 40
5.12 × 1 = 12 117 - 12 = 105 40
6.12 × (-1) = -12 105 - (-12) = 117 40
7.12 × 1 = 12 117 - 12 = 105 40
8.12 × (-1) = -12 105 - (-12) = 117 40
9.12 × 1 = 12 117 - 12 = 105 4
10.12 × (-1) = -12 105 - (-12) = 117 41
1.12 × 1 = 12 117 - 12 = 105 41
2.12 × (-1) = -12 105 - (-12) = 117 41
3.12 × 1 = 12 117 - 12 = 105 41
4.12 × (-1) = -12 105 - (-12) = 117 41
5.12 × 1 = 12 117 - 12 = 105 41
6.12 × (-1) = -12 105 - (-12) = 117 41
7.12 × 1 = 12 117 - 12 = 105 41
8.12 × (-1) = -12 105 - (-12) = 117 41
9.12 × 1 = 12 117 - 12 = 105 420. 12 × (-1) = -12 105 - (-12) = 117 42
1.12 × 1 = 12 117 - 12 = 105 42
2.12 × (-1) = -12 105 - (-12) = 117 42
3.12 × 1 = 12 117 - 12 = 105 42
4.12 × (-1) = -12 105 - (-12) = 117 42
5.12 × 1 = 12 117 - 12 = 105 42
6.12 × (-1) = -12 105 - (-12) = 117 42
7.12 × 1 = 12 117 - 12 = 105 42
8.12 × (-1) = -12 105 - (-12) = 117 42
9.12 × 1 = 12 117 - 12 = 105 430. 12 × (-1) = -12 105 - (-12) = 117 43
1.12 × 1 = 12 117 - 12 = 105 43
2.12 × (-1) = -12 105 - (-12) = 117 43
3.12 × 1 = 12 117 - 12 = 105 43
4.12 × (-1) = -12 105 - (-12) = 117 43
5.12 × 1 = 12 117 - 12 = 105 43
6.12 × (-1) = -12 105 - (-12) = 117 43
7.12 × 1 = 12 117 - 12 = 105 43
8.12 × (-1) = -12 105 - (-12) = 117 43
9.12 × 1 = 12 117 - 12 = 105 440. 12 × (-1) = -12 105 - (-12) = 117 44
1.12 × 1 = 12 117 - 12 = 105 44
2.12 × (-1) = -12 105 - (-12) = 117 44
3.12 × 1 = 12 117 - 12 = 105 44
4.12 × (-1) = -12 105 - (-12) = 117 44
5.12 × 1 = 12 117 - 12 = 105 44
6.12 × (-1) = -12 105 - (-12) = 117 44
7.12 × 1 = 12 117 - 12 = 105 44
8.12 × (-1) = -12 105 - (-12) = 117 44
9.12 × 1 = 12 117 - 12 = 105 450. 12 × (-1) = -12 105 - (-12) = 117 45
1.12 × 1 = 12 117 - 12 = 105 45
2.12 × (-1) = -12 105 - (-12) = 117 45
3.12 × 1 = 12 117 - 12 = 105 45
4.12 × (-1) = -12 105 - (-12) = 117 45
5.12 × 1 = 12 117 - 12 = 105 45
6.12 × (-1) = -12 105 - (-12) = 117 45
7.12 × 1 = 12 117 - 12 = 105 45
8.12 × (-1) = -12 105 - (-12) = 117 45
9.12 × 1 = 12 117 - 12 = 105 460. 12 × (-1) = -12 105 - (-12) = 117 46
1.12 × 1 = 12 117 - 12 = 105 46
2.12 × (-1) = -12 105 - (-12) = 117 46
3.12 × 1 = 12 117 - 12 = 105 46
4.12 × (-1) = -12 105 - (-12) = 117 46
5.12 × 1 = 12 117 - 12 = 105 46
6.12 × (-1) = -12 105 - (-12) = 117 46
7.12 × 1 = 12 117 - 12 = 105 46
8.12 × (-1) = -12 105 - (-12) = 117 46
9.12 × 1 = 12 117 - 12 = 105 470. 12 × (-1) = -12 105 - (-12) = 117 47
1.12 × 1 = 12 117 - 12 = 105 47
2.12 × (-1) = -12 105 - (-12) = 117 47
3.12 × 1 = 12 117 - 12 = 105 47
4.12 × (-1) = -12 105 - (-12) = 117 47
5.12 × 1 = 12 117 - 12 = 105 47
6.12 × (-1) = -12 105 - (-12) = 117 47
7.12 × 1 = 12 117 - 12 = 105 47
8.12 × (-1) = -12 105 - (-12) = 117 47
9.12 × 1 = 12 117 - 12 = 105 480. 12 × (-1) = -12 105 - (-12) = 117 48
1.12 × 1 = 12 117 - 12 = 105 48
2.12 × (-1) = -12 105 - (-12) = 117 48
3.12 × 1 = 12 117 - 12 = 105 48
4.12 × (-1) = -12 105 - (-12) = 117 48
5.12 × 1 = 12 117 - 12 = 105 48
6.12 × (-1) = -12 105 - (-12) = 117 48
7.12 × 1 = 12 117 - 12 = 105 48
8.12 × (-1) = -12 105 - (-12) = 117 48
9.12 × 1 = 12 117 - 12 = 105 490. 12 × (-1) = -12 105 - (-12) = 117 49
1.12 × 1 = 12 117 - 12 = 105 49
2.12 × (-1) = -12 105 - (-12) = 117 49
3.12 × 1 = 12 117 - 12 = 105 49
4.12 × (-1) = -12 105 - (-12) = 117 49
5.12 × 1 = 12 117 - 12 = 105 49
6.12 × (-1) = -12 105 - (-12) = 117 49
7.12 × 1 = 12 117 - 12 = 105 49
8.12 × (-1) = -12 105 - (-12) = 117 49
9.12 × 1 = 12 117 - 12 = 105 500. 12 × (-1) = -12 105 - (-12) = 117 50
1.12 × 1 = 12 117 - 12 = 105 50
2.12 × (-1) = -12 105 - (-12) = 117 50
3.12 × 1 = 12 117 - 12 = 105 50
4.12 × (-1) = -12 105 - (-12) = 117 50
5.12 × 1 = 12 117 - 12 = 105 50
6.12 × (-1) = -12 105 - (-12) = 117 50
7.12 × 1 = 12 117 - 12 = 105 50
8.12 × (-1) = -12 105 - (-12) = 117 50
9.12 × 1 = 12 117 - 12 = 105 5
10.12 × (-1) = -12 105 - (-12) = 117 51
1.12 × 1 = 12 117 - 12 = 105 51
2.12 × (-1) = -12 105 - (-12) = 117 51
3.12 × 1 = 12 117 - 12 = 105 51
4.12 × (-1) = -12 105 - (-12) = 117 51
5.12 × 1 = 12 117 - 12 = 105 51
6.12 × (-1) = -12 105 - (-12) = 117 51
7.12 × 1 = 12 117 - 12 = 105 51
8.12 × (-1) = -12 105 - (-12) = 117 51
9.12 × 1 = 12 117 - 12 = 105 520. 12 × (-1) = -12 105 - (-12) = 117 52
1.12 × 1 = 12 117 - 12 = 105 52
2.12 × (-1) = -12 105 - (-12) = 117 52
3.12 × 1 = 12 117 - 12 = 105 52
4.12 × (-1) = -12 105 - (-12) = 117 52
5.12 × 1 = 12 117 - 12 = 105 52
6.12 × (-1) = -12 105 - (-12) = 117 52
7.12 × 1 = 12 117 - 12 = 105 52
8.12 × (-1) = -12 105 - (-12) = 117 52
9.12 × 1 = 12 117 - 12 = 105 530. 12 × (-1) = -12 105 - (-12) = 117 53
1.12 × 1 = 12 117 - 12 = 105 53
2.12 × (-1) = -12 105 - (-12) = 117 53
3.12 × 1 = 12 117 - 12 = 105 53
4.12 × (-1) = -12 105 - (-12) = 117 53
5.12 × 1 = 12 117 - 12 = 105 53
6.12 × (-1) = -12 105 - (-12) = 117 53
7.12 × 1 = 12 117 - 12 = 105 53
8.12 × (-1) = -12 105 - (-12) = 117 53
9.12 × 1 = 12 117 - 12 = 105 540. 12 × (-1) = -12 105 - (-12) = 117 54
1.12 × 1 = 12 117 - 12 = 105 54
2.12 × (-1) = -12 105 - (-12) = 117 54
3.12 × 1 = 12 117 - 12 = 105 54
4.12 × (-1) = -12 105 - (-12) = 117 54
5.12 × 1 = 12 117 - 12 = 105 54
6.12 × (-1) = -12 105 - (-12) = 117 54
7.12 × 1 = 12 117 - 12 = 105 54
8.12 × (-1) = -12 105 - (-12) = 117 54
9.12 × 1 = 12 117 - 12 = 105 550. 12 × (-1) = -12 105 - (-12) = 117 55
1.12 × 1 = 12 117 - 12 = 105 55
2.12 × (-1) = -12 105 - (-12) = 117 55
3.12 × 1 = 12 117 - 12 = 105 55
4.12 × (-1) = -12 105 - (-12) = 117 55
5.12 × 1 = 12 117 - 12 = 105 55
6.12 × (-1) = -12 105 - (-12) = 117 55
7.12 × 1 = 12 117 - 12 = 105 55
8.12 × (-1) = -12 105 - (-12) = 117 55
9.12 × 1 = 12 117 - 12 = 105 560. 12 × (-1) = -12 105 - (-12) = 117 56
1.12 × 1 = 12 117 - 12 = 105 56
2.12 × (-1) = -12 105 - (-12) = 117 56
3.12 × 1 = 12 117 - 12 = 105 56
4.12 × (-1) = -12 105 - (-12) = 117 56
5.12 × 1 = 12 117 - 12 = 105 56
6.12 × (-1) = -12 105 - (-12) = 117 56
7.12 × 1 = 12 117 - 12 = 105 56
8.12 × (-1) = -12 105 - (-12) = 117 56
9.12 × 1 = 12 117 - 12 = 105 570. 12 × (-1) = -12 105 - (-12) = 117 57
1.12 × 1 = 12 117 - 12 = 105 57
2.12 × (-1) = -12 105 - (-12) = 117 57
3.12 × 1 = 12 117 - 12 = 105 57
4.12 × (-1) = -12 105 - (-12) = 117 57
5.12 × 1 = 12 117 - 12 = 105 57
6.12 × (-1) = -12 105 - (-12) = 117 57
7.12 × 1 = 12 117 - 12 = 105 57
8.12 × (-1) = -12 105 - (-12) = 117 57
9.12 × 1 = 12 117 - 12 = 105 580. 12 × (-1) = -12 105 - (-12) = 117 58
1.12 × 1 = 12 117 - 12 = 105 58
2.12 × (-1) = -12 105 - (-12) = 117 58
3.12 × 1 = 12 117 - 12 = 105 58
4.12 × (-1) = -12 105 - (-12) = 117 58
5.12 × 1 = 12 117 - 12 = 105 58
6.12 × (-1) = -12 105 - (-12) = 117 58
7.12 × 1 = 12 117 - 12 = 105 58
8.12 × (-1) = -12 105 - (-12) = 117 58
9.12 × 1 = 12 117 - 12 = 105 590. 12 × (-1) = -12 105 - (-12) = 117 59
1.12 × 1 = 12 117 - 12 = 105 59
2.12 × (-1) = -12 105 - (-12) = 117 59
3.12 × 1 = 12 117 - 12 = 105 59
4.12 × (-1) = -12 105 - (-12) = 117 59
5.12 × 1 = 12 117 - 12 = 105 59
6.12 × (-1) = -12 105 - (-12) = 117 59
7.12 × 1 = 12 117 - 12 = 105 59
8.12 × (-1) = -12 105 - (-12) = 117 59
9.12 × 1 = 12 117 - 12 = 105 600. 12 × (-1) = -12 105 - (-12) = 117 60
1.12 × 1 = 12 117 - 12 = 105 60
2.12 × (-1) = -12 105 - (-12) = 117 60
3.12 × 1 = 12 117 - 12 = 105 60
4.12 × (-1) = -12 105 - (-12) = 117 60
5.12 × 1 = 12 117 - 12 = 105 60
6.12 × (-1) = -12 105 - (-12) = 117 60
7.12 × 1 = 12 117 - 12 = 105 60
8.12 × (-1) = -12 105 - (-12) = 117 60
9.12 × 1 = 12 117 - 12 = 105 6
10.12 × (-1) = -12 105 - (-12) = 117 61
1.12 × 1 = 12 117 - 12 = 105 61
2.12 × (-1) = -12 105 - (-12) = 117 61
3.12 × 1 = 12 117 - 12 = 105 61
4.12 × (-1) = -12 105 - (-12) = 117 61
5.12 × 1 = 12 117 - 12 = 105 61
6.12 × (-1) = -12 105 - (-12) = 117 61
7.12