Theory: Number Theory (1): Divisible, gcd, and LCM

The nature of divisible Division

Set A, B is two integers, and b≠0. If there is an integer c, so that a = b * C, it is said that A is divisible by B, or B is divisible by a, as b |a(here is a is divisible by B, a >= b)
At this time also called A is a multiple of B, B is a factor. If B does not divide a, remember

• Divide the basic definition

Definition 1.1: If n is 2 in addition to the remainder of 0, then for an integer k, there is n = 2k, we call n is an even number, and if the remainder of n by 2 is 1, we have n = 2k + 1 for an integer k, we call n an odd number.
definition 1.2: Set A, B is two integers, and b≠0, there is a unique integer q and R, which makes:
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This expression is called the band-rest division and is recorded as R = A (mod b); for example -13 mod 5 = 3

• The nature of divisible

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Application of divisible

the number of digits divisible
Fibonacci divisible

···· GCD and LCM Basic definitions

Definition 1.3: Set A and B are two integers, if d|a, and d|b, then we call D a common factor of A and B.
definition 1.4: Set A and B are two integers that are not all 0, the largest of the common factors of A and B is the largest common factor of a and B, or greatest common divisor, which is recorded as GCD (A, b), sometimes précis-writers (A, B)
Definition 1.5: Set A and B are two non-0 certificates, the minimum positive common multiple of a and B are least common multiple of A and B, recorded as LCM (A, b), sometimes précis-writers to [A, b]

Basic Operation Properties

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About the operations of GCD and LCM

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The above two figures are the legendary Euclidean theorem.

Proof of the nature of GCD operation

Brief proof:

We say that the recursion is true because any common factor of M and N must also be the male factor of M and (n mod m).
Proof that the above sentence is based on the following formula
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Here we can introduce the Euclidean theorem of expansion.

Full proof:

where (A,b,q,r) ∈z is set, then:
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！ Certificate: Only A and B, B and r have the same common factor.
(1). Set D is the common factor of A and B, i.e. (D|a) and (d|b). We notice that
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So that we can get the following derivation process:
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Here we prove the positive determination in the case of a|b (that is, gcd (0,n) is correct in the expression above)
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(2). Set D is the common factor of B and r, so we have the following derivation:
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Pseudo-code description of GCD and LCM algorithms

while(max0)    min→temp    maxmodmin→min    temp→maxreturn <max>

Examples

Two-Instrument Sword

Theory: Number Theory (1): Divisible, gcd, and LCM