# "Algorithm" greatest common divisor, least common multiple, mathematical induction method

Source: Internet
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Greatest common divisor

If a number a can be divisible by a number B, a is called a multiple of B, and B is called an approximate.

The number of public approximations in several integers, called these number of conventions, the largest of which is called the greatest common divisor of these few numbers.

12, 16 of the conventions are 1, 2, 4, the largest of which is the greatest common divisor of 12 and 16, generally recorded as (12,16) = 4.

The purpose of the number of conventions is numerator:

Dividing the numerator and denominator of a fraction by their number of conventions, the value of the fraction is constant, and the process is called numerator;

Numerator makes this score easier to use.

Least common multiple

Several natural numbers of public multiples, called these numbers of common multiple, of which the smallest of a natural number, called these several numbers of least common multiple.

Multiples of 4 have 4, 8, 12......,6 multiples of 6, 12, 18......,4 and 6 common multiple have 12, 24, ..., the smallest of which is 12.

Generally recorded as [4,6]=12.

The purpose of common multiple is-pass:

The process of fractional differentiation of several different denominators into the same denominator as the original fraction, called-pass.

If you want to add and subtract two fractions, it's best to make him the same two points in the denominator for easy calculation.

At this point you can find the least common multiple of the denominator of the two scores, and then there is a way to do it.

Mathematical inductive method is a mathematical proof method, which is often used to prove that a given proposition is set up in the whole (or part) natural number range. In addition to the natural number, the generalized mathematical induction can also be used to prove the general LIANGJI structure, for example: The tree in set theory. This generalized mathematical induction method is applied to the field of mathematical logic and computer science, which is called the structure induction method.
In number theory, mathematical induction is a mathematical theorem that proves that any given situation is correct in a different way (the first, the second, the third, and so on, without exception).
Although the mathematical induction of the name of "induction", but the mathematical induction is not a rigorous inductive reasoning method, it belongs to a completely rigorous deductive reasoning method. In fact, all mathematical proofs are deductive.
The simplest and most common mathematical induction method is to prove that a proposition is established when n equals any natural number. The proof is divided into the following two steps:
1. Prove that when n= 1 o'clock proposition is set up.
2. Assuming n=m when the proposition is established, it can be deduced that the proposition is also set up in N=m+1. (m stands for any natural number)
The principle of this approach is to first prove that the proposition is set at a certain starting point value, and then prove that the process from one value to the next value is valid. When these two points have been proved, then any value can be deduced by repeated use of this method.
Here's a math problem: an example proves the following theorem: The first step is to verify that the formula is set at n = 1 o'clock. That is, left = 1, right so this formula is set at n = 1 o'clock. The second step, we need to prove that n = m when the formula is established, then we can deduce n = m+1 when the formula is also established. The steps are as follows: assuming n = m when the formula is true (Equation 1) and then adding M + 1 at the same time on both sides of the equation (equation 2) This is the equation for n = m+1. Our next step is to prove that equation 2 is established according to equation 1. By factoring the combination, the right side of Equation 2: That is: So we have completed the establishment of N=M+1 by the N=M established process, proof. Conclusion: For any natural number n, the formula is established.

"Algorithm" greatest common divisor, least common multiple, mathematical induction method

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