Number Systems Shortcuts
Product of any two numbers = Product of their HCF and LCM
For three positive numbers a, b , c
(a + b + c) * (1/a + 1/b + 1/c) >= 9
For any positive integer n
2 <= (1 + 1/n)n <= 3
a2+b2+c2 >= ab + bc + ca
If a = b = c , then a2+b2+c2 = ab + bc + ca
a4+b4+c4+d4 >= 4abcd
If a + b + c + d = const, then the ap * bq * cr * ds will be maximum if a/p = b/q = c/r = d/s .
(n!)2 > nn (! for factorial)
If n is even, n(n+1)(n+2) is divisible by 24
an - bn is always divisible by a - b
Very useful incase of remainder problems.
For example 173 - 143 will be completely divisible by 3 ( = 17 - 14 )
Product of any two numbers = Product of their HCF and LCM
For three positive numbers a, b , c
(a + b + c) * (1/a + 1/b + 1/c) >= 9
For any positive integer n
2 <= (1 + 1/n)n <= 3
a2+b2+c2 >= ab + bc + ca
If a = b = c , then a2+b2+c2 = ab + bc + ca
a4+b4+c4+d4 >= 4abcd
If a + b + c + d = const, then the ap * bq * cr * ds will be maximum if a/p = b/q = c/r = d/s .
(n!)2 > nn (! for factorial)
If n is even, n(n+1)(n+2) is divisible by 24
an - bn is always divisible by a - b
Very useful incase of remainder problems.
For example 173 - 143 will be completely divisible by 3 ( = 17 - 14 )
(m+n)! is divisible by m! * n!
2 <= (1+1/n)n <= 3 (1 + x)n ~ (1 + nx) if x <<< 1 when a three digit number is reversed and the difference of these two numbers is taken , the middle number is always 9 and the sum of the other two numbers is always 9 . 22n - 1 is always divisible by 3 If for two numbers x + y = constant ( say k), then their product is maximum if x = y (= k/2). The maximum product is then (k2)/4 If for two numbers x * y = constant( say k), then their sum is minimum if x = y = √k. The minimum sum is then 2√k When a number can be expressed as a product of n distinct primes, then it can be expressed as a product of 3 numbers in (3(n+1) + 1)/2 ways if a + b + c = 0 then a3 + b3 + c3 = 3abc |
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For a(P - 1) / P where P is a prime, the remainder would be 1 (Fermat theorm)
For (P-1)!/P where P is a prime, the remainder would be P - 1 (Wilson's theorm)
Euler’s totient function, E[n] denotes number of positive integers that are coprime to and less than a certain positive integer n. For a positive integer N having prime divisors p1, p2, ... , pn; E[N] = N (1 - 1/p1) (1 - 1/p2)... (1 - 1/pn). Euler's theorem states that if P and N are positive coprime integers, then when PE[N]/ N will give the remainder as 1. ( Euler's theorm)
For (P-1)!/P where P is a prime, the remainder would be P - 1 (Wilson's theorm)
Euler’s totient function, E[n] denotes number of positive integers that are coprime to and less than a certain positive integer n. For a positive integer N having prime divisors p1, p2, ... , pn; E[N] = N (1 - 1/p1) (1 - 1/p2)... (1 - 1/pn). Euler's theorem states that if P and N are positive coprime integers, then when PE[N]/ N will give the remainder as 1. ( Euler's theorm)