E-StoreWant to Know Mathematical Facts, Theorems to Prepare Best for Entrance?
These are some pretty good mathematical fact...
Bertrand’s Postulate: For every positive integer n, there exists a prime
p such that p lies in [n,2n].
Gauss’s Lemma: If a polynomial with integer coefficients factors into
two polynomials with rational coefficients, then it factors into two polynomials
with integer coefficients.
Modular Arithmetic:
For a positive integer m and integers a and b, we say that a is congruent to
b modulo m if m | (a − b), and we denote this by a = b modulo m, or more
commonly a = b (mod m). Otherwise, a is not congruent to b modulo m,
and we denote this by a 6= b (mod m) (although this notation is not used
often). In the above notation, m is called the modulus, and we consider the
integers modulo m.
Theorem. If a = b and c = d (mod m), then a+c = b+d (mod m)
and ac = bd (mod m).
If p is a prime and p = 1 or 3 (mod 8), then there exist positive
integers x and y such that p = x2 + 2y2.
If p is a prime and p = 1 (mod 6), then there exist positive integers
x and y such that p = x2 + 3y2
If f is a polynomial with integer coefficients and a = b (mod m), then
f(a) = f(b) (mod m).
• If f is a polynomial with integer coefficients of degree n, not identically
zero, and p is a prime, then the congruence
f(x) = 0 (mod p) has at most n solutions modulo p, counting multiplicity.
Euler’s Theorem: If a is relatively prime to m, then
a t(m) = (mod m) where t(m) is teh totient function.
a number p is prime iff (p-1)! + 1 is divisible by p.
also p is prime if t(p) = p -1 where t(p) denotes the totient function.
for any integer n , np - n is divisible by p if p is prime. if the GCD of n and p is 1.
Some of the Related Articles:
Tough Trigonometric Questions .. Do Practice with these !!
Practice Questions of IUPAC Nomenclature
Practice Questions on Permutation & Combination
Questions of Inequality in mathematics
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Ya very very interesting. You only study maths??? No physics?? And no chemistry???