Elementary Number Theory Study Note one: The concept of divisible Division and band remainder

The definition of divisible: A, B is two arbitrary integers, b!=0, if there is an integer q, so that A=b*q, said "A can be divisible by B", or "B can be divisible by a", or "B is a factor", or "A is a multiple of B", with the notation "B|a".

The nature of the divisible:
(1) b|0, because 0=b*0
(2) 1|a, because a=1*a
(3) B|a <=> b| ABS (a)
(4) a|b, b|c = A|c, because a|b,b|c = B=A*Q1,C=B*Q2 = c=a*q1*q2

The theorem 1:a is the same as that of ABS (a).
Because 6=1*6=2*3= (-1) * (-6) = (-2) * (-3), the factor of 6 is 1,-1,2,-2,3,-3,6,-6.
All the constraints of + A and abs (a) are the same because a=b*q <=> a = (-B) * (-Q).
The approximations of a and a are in pairs, b|a <=>-b|a, because the factors are in pairs, so only positive factors are discussed in the usual discussion.

Theorem 2:a,b are integers, which are multiples of M, then a+b,a-b are multiples of M.

Theorem 3: If A1,a2,..., An is a multiple of M, then Q1*a1+q2*a2+...+qn*an is a multiple of M.

Theorem 4 (with remainder division): If A, B is a two integer, b>0, there are two integer q,r, which makes
A=b*q+r
Among them 0<=r<b, and, Q,r is the only.
such as: When A=16,b=5,16=5*3+1
When a=-16,b=5,-16=5* ( -3) -1=6* ( -3) +2 = 16 divided by 5 the remainder is 2
When a=20,b=6,20=6*3+2
When a=-20,b=6,-20=6* ( -3) -2=6* ( -4) +4 = 20 divided by 6 the remainder is 4
A= B*q+r
-a= b* (-Q)-R
(when r=0) = b* (-Q)
(when r!=0) = b* (-q-1) +b-r
Note: Running -16/5=-3,-16%5=-1 in the GCC compiler of MINGW is not the same as the theory.
Proof: Do integer series ..., -3b,-2b,-b,0,b,2b,3b,...
Presence Q makes q*b<=a< (q+1) *b
Make r=a-q*b, then A=q*b+r
r>=0,q-q*b-b<0 = A-q*b<b = R<b
= 0<=r<b
Set A=B*Q1+R1, 0<=r1<b
B (q-q1) +r-r1=0,b (Q-Q1) =r1-r
B*abs (Q-Q1) =abs (R-R1) <b, if Q-q1!=0,b*abs (q-q1) >=b = ABS (R-R1) >=b
And because 0<=r<b =-b<r-r1<b = ABS (R-R1) <b
= = r = R1

Inference: If a, B is a two integer, there are two integer q,r, which makes
A=b*q+r
where 0<=r<abs (b), and Q,r is unique.

With the remainder division A=b*q+r,0<=r<b, Q is called a divided by B of the incomplete quotient, R is called a divided by B.

Elementary Number Theory Study Note one: The concept of divisible Division and band remainder