(a) the basic law of parity operation odd plus or minus odd = even; Even number = even number; Even odd number = odd number; Odd even number = odd number. Inference 1. If the sum of any two numbers is odd, then the difference is also odd; If the sum is even, then the difference is even. 2. If the sum or difference of any two numbers is odd, the parity of the two numbers is opposite; If the sum or difference is an even number, the parity of the two numbers is the same. (2) The basic rule of divisibility judgment is 1. Numerical characteristics of numbers divisible by 2, 4, 8, 5, 25, 125-numbers divisible by 2 (or 5), with the last digit divisible by 2 (or 5); A number divisible by 4 (or 25), and the last two digits can be divisible by 4 (or 25); Numbers divisible by 8 (or 125), numbers divisible by 8 in the last three digits (or125); The remainder of a number divided by 2 (or 5) is the remainder of its last number divided by 2 (or 5); The remainder of a number divided by 4 (or 25) is the remainder of its last two digits divided by 4 (or 25); The remainder of a number divided by 8 (or 125) is the remainder of its last three digits divided by 8 (or 125). 2. Numeric characteristics of numbers divisible by 3 and 9-the sum of numbers divisible by 3 (or 9) and numbers divisible by 3 (or 9). The remainder of a number divided by 3 (or 9) is the remainder of the number divided by 3 (or 9) after adding the bits. 3. Numerical characteristics of numbers divisible by 1 1-Numbers divisible by 1 1, and the difference between the sum of odd numbers and the sum of even numbers can be divisible by 1 1. (3) The core judgment feature of multiple relation If a: b = m: n (m, n is coprime), then a is a multiple of m; B is a multiple of n, if NX = my (m, n is coprime), then X is a multiple of m; Y is a multiple of n, and if a∶b=m∶n(m, n is coprime), then a b should be a multiple of m n. The formula of multiplication and factorization is the law of forward multiplication distribution: (a+b) c = AC+BC; Inverse multiplication distribution law: AC+BC = (a+b) c; (also called "common factor extraction method") Variance: A2-B2 = (a-b) (a+b); Complete sum/difference of squares: (ab) 2 = A2AB+B2; Cubic sum: A3+B3 = (a+b) (A2-AB+B2); Cubic difference: A3-B3 = (a-b) (A2+AB+B2); Complete cubic sum/difference: (A b) 3 = A33 (A2) B+3A (B2) B3; Sum formula of proportional series: s = a1(1-qn)/(1-q) (q ≠1); Sum formula of arithmetic sequence: Sn=na 1+n(n- 1)d/2 or Sn=n(a 1+an)/2. Trigonometric inequality 买 a+b 买 a 买+买 b 买; 丨丨丨 A 丨+丨丨 B 丨; 丨丨 a-b 丨≥丨A-B 丨; -a, a, a, a, a, a, a; 丨丨 b = >-b ≤ a ≤ b. The sum of the first n items in some series is1+2+3+…+n = n (n+1)/2; 1+3+5+…+(2n- 1)= N2； 2+4+6+…+(2n)= n(n+ 1)； 12+32+52+…+(2n- 1)2 = n(4 N2- 1)/3 13+33+…+n3 =(n+ 1)2 N2/4 13+33+53+…+(2n- 1)3 = N2(2 N2- 1)65438