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Testing Techniques: Chinese Remainder Theorem

2019-12-26 10:10:35 | Source: Hao Ling, CPG Education

Candidates, many students encountered the problem of Chinese remaining theorem in the process of preparing for the exam. In addition to substituting and eliminating this method, they were at a loss. In fact, it is easier to master the problem of remaining theorem in China. Let ’s study this part together.

What is the Chinese Remainder Theorem? The Chinese Remainder Theorem first appeared in the "Sun Tzu's Sutra", also known as "the matter of unknown number". The remaining two of the seven number, the matter geometry? That is, an integer divided by three remaining two, divided by five remaining three, divided by seven remaining two, to find this integer. The Sun Tzu's Arithmetic mentioned the problem of congruence equations for the first time, as well as the solutions to the above specific problems. Therefore, the Chinese remainder theorem is also called the Sun Tzu theorem in the Chinese mathematics literature. The general form of the Chinese remainder theorem is: divide M by A to get the remainder a; divide by B to get the remainder b; divide M by C to get the remainder c; how much is M? There are some special models as follows:

1. Identical addition and remainder, for example: M ÷ 3 ... 1, M ÷ 4 ... 1, then M = 12n + 1

Let's look at an example:

Example 1. A positive integer greater than 10, divided by 3 remainders 2, divided by 4 remainders 2, and divided by 5 remainders 2. Ask what is the minimum number?

A.60 B.61 C.62 D.63

[Answer] C. Chinese public analysis: Divide a number M by A to get the remainder a; divide by B to get the remainder b; divide by C to get the remainder c. Find the form of this number, which conforms to the Chinese remainder theorem. And the remainders are 2, which conforms to the model of congruence and remainder. The appropriate number in this question should be the common multiples of 3, 4, and 5 plus 2, all such numbers can be expressed as 60n + 2 (n is an integer), because this number is greater than 10, when n is 1, this The minimum number is 62. Choose C.

Second, the difference is the same as the difference, for example: M ÷ 5 ... 2, M ÷ 4 ... 1, then M = 20n-3

Let's look at an example:

Example 2. A positive integer P less than 200 is divided by 11 and 8 and divided by 13 and 10. What is P?

A.139 B.140 C.141 D.142

[Answer] B. Cong Gong analysis: This topic is a number less than two hundred divided by 11 and 8 and divided by 13 and 10. Finding this number is in accordance with the Chinese remainder theorem. 11-8 = 3, 13-10 = 3, the difference between the divisor and the remainder is 3, and the least common multiple of 11, 13 is 143. According to the difference and subtraction, P = 143n-3, then the number less than 200 In this case, the value of P is 140. Choose B.

Third, and add together, for example: M ÷ 3 ... 2, M ÷ 4 ... 1, then M = 12n + 5

Let's look at an example:

Example 3. A number more than a hundred, divided by 9 more than 2 and divided by 8 more than 3, what is this number?

A.153 B.154 C.154 D.155

[Answer] D. Chinese public analysis: According to the explanation above, you can determine that this question conforms to the form of the Chinese remainder theorem, because 9 + 2 = 11, 8 + 3 = 11, the sum of the divisor and the remainder is 11, and the least common multiple of 8, 9 is 72. According to sum and same addition, it can be known that the dividend can be expressed as 72n + 11, and it is known that the dividend is greater than 100 and less than 200, so n = 2, this number is 155. Choose D.

In the special model of the Chinese Remainder Theorem, there are surplus-plus-plus, difference-plus-minus, and plus-plus. Everyone prepares for the test by digesting it with more questions, so that they can be solved smoothly in the exam.

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(Responsible editor: Zhang Ye)

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