# Formulas and Theories: Numbers

(1) If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if
x=y(=k/2). The maximum product is then (k^2)/4 .

(2) If for two numbers x*y=k(=constant), then their SUM is MINIMUM if
x=y(=root(k)). The minimum sum is then 2*root(k) .

(3) |x| + |y| >= |x+y| (|| stands for absolute value or modulus )
(Useful in solving some inequations)

(4) Product of any two numbers = Product of their HCF and LCM .
Hence product of two numbers = LCM of the numbers if they are prime to each other .

6) For any 2 numbers a>b

a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa respectively)

(7) (GM)^2 = AM * HM

(8) For three positive numbers a, b ,c

(a+b+c) * (1/a+1/b+1/c)>=9

(9) For any positive integer n

2<= (1+1/n)^n <=3

(10) a^2+b^2+c^2 >= ab+bc+ca, If a=b=c , then the equality holds in the above.

(11) a^4+b^4+c^4+d^4 >=4abcd

(12) (n!)^2 > n^n (! for factorial)

(13) (m+n)! is divisible by m! * n! .

(14)  2<= (1+1/n)^n <=3

(15) (1+x)^n ~ (1+nx) if x<<<1

(16) Let ‘x’ be certain base in which the representation of a number is ‘abcd’ , then the decimal value of this number is a*x^3 + b*x^2 + c*x + d