Rolling and dividing method:
If d is the common factor of a and b, that is, d|a, d|b, then d|a-bq=r, thus d is the common factor of b and r, and it can be proved in the same way that the common factors of r and b are equal is the common factor of a and b, therefore, the set of common factors of a and b is the same as the set of common factors of r and b, so their greatest common factors are equal, that is, gcd(a, b)=gcd(b, r).
Important properties of the greatest common factor:
Suppose the integers a and b are not zero at the same time, then there is a pair of integers m and n such that (a, b)=am+bn.
若gcd(a，n)=1，gcd(b，n)=1，则gcd(ab，n)=1.
gcd(ma，mb)=gcd(a，b)×m
An important property of divisibility:
If a|bc, and gcd(a, b)=1, then a|c.
An important property of prime numbers:
Let p be a prime number, if p|ab, then p|a, or p|b.
The relationship between the greatest common factor and the greatest common multiple:
gcd(a，b)lcm(a，b)=|ab|.
gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c))
lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c))
Fundamental Theorem of Arithmetic:
Any integer greater than 1 can always be decomposed into the form of the product of prime factors, and, regardless of the order of the prime factors in the decomposition formula, this decomposition formula is unique.
The relationship between congruence and divisibility:
a≡b(mod n) is equivalent to n|ab.
Congruent properties:
1. If a≡b (mod m), b≡c (mod m), then a≡c (mod m)
2. If a≡b(mod n), and c≡d(mod n), then
- a+c≡b+d(mod n)；
- ac≡bd(mod n)；
- ka≡kb(mod n), k is any integer;
- am≡bm(mod n), m is a positive integer;
3、a*b mod k = (a mod k) * (b mod k) mod k
4. If a≡b(mod p), a≡b(mod q), p, q are mutually prime, then a≡b(mod p*q).
5. If ab≡ac(mod n), and gcd(a, n)=1, then b≡c(mod n).
The necessary and sufficient conditions for non-zero elements [a] to have inverse elements:
gcd(a，n)=1.
Fermat's little theorem:
Let m be a prime number, a be any integer, and gcd(a, m)=1, then a^m-1≡1(mod m)
Euler's theorem:
Let m be a positive integer, a be any integer, and gcd (a, m)=1, then a^Φ(m)≡1(mod m)
Regarding the complete residual system and simplified residual system modulo n, the following basic conclusions are obtained:
Theorem 1: Suppose n is a positive integer, a, b are integers and gcd (a, n)=1, if {a1, a2, ..., an} is a complete residual system modulo n, then {aa1+b, aa2 +b,...,aan+b} is also a complete residual system modulo n.
Theorem 2: Suppose n is a positive integer, a is an integer and gcd (a, n)=1, if {a1, a2, ..., aΦ(n)} is a simplified residual system modulo n, then {aa1, aa2, ..., aaΦ(n) } is also a reduced remainder system modulo n.
Theorem 3: Suppose m, n are positive integers and gcd(m, n)=1, if {a1, a2,…,an} is a complete residual system modulo m, {b1, b2,…,bn} is modulo A complete residual system of n, then the set of all integers nai+mbj is a complete residual system of mn.
Theorem 4: Suppose m, n are positive integers and gcd(m, n)=1, if {a1, a2, ..., an} is a simplified residual system modulo m, {b1, b2, ..., bn} is modulo A simplified residual system of n, then the set of all integers nai+mbj is a simplified residual system of mn. The relationship can be obtained: if m, n are positive integers, and (m, n)=1, then Φ(mn )=Φ(m)Φ(n).
Theorem 5: Let n be an integer greater than 1, then there is the following expression:
Φ(n)=n(1-1/p1)(1-1/p2)…(1-1/pk)，
Among them, p1, p2, ..., pk are all distinct prime factors of n.
Congruence equation conclusion:
The first-order congruence equation a≡b(mod n) has a solution, then gcd (a, n)|b. Conversely, when gcd (a, n)|b, the first-order congruence equation a≡b(mod n) has exactly (a, n) solutions.
A criterion for determining that the indeterminate equation ax+by=c has integer solutions:
If the indeterminate equation has integer solutions, then gcd (a,b)|c. Conversely, when gcd (a, b)|c, the indeterminate equation must have integer solutions.
Conclusions about the solution of the indeterminate equation ax+by=c:
Let gcd(a, b)=1, then the integer general solution of the indeterminate equation ax+by=c is:
x=x0+bt
y=y0-at
Where t is any integer, x=x0, y=y0 is a special solution of the indeterminate equation ax+by=c.