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#### Odd and even numbers , Polynomials with integer coefficients and integer values , Rational and irrational numbers

Are there such irrational numbers a and b so that a $>$ 1, b $>$ 1, and [$a^m$] is different from [$b^n$] for any natural numbers m and n?

#### Odd and even numbers , Polynomials with integer coefficients and integer values , Rational and irrational numbers

n numbers are given as well as their product, p. The difference between p and each of these numbers is an odd number.
Prove that all n numbers are irrational.

#### Polynomials with integer coefficients and integer values , Rational and irrational numbers , The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

Prove that if $(p, q) = 1$ and p/q is a rational root of the polynomial $P (x) = a_nx^n + … + a_1x + a_0$ with integer coefficients, then

a) $a_0$ is divisible by p;

b) $a_n$ is divisible by q.

#### Extremal principle (other) , Mathematical induction (other) , Polynomial remainder theorem (little Bezout's theorem). Factorisation. , Polynomials with integer coefficients and integer values , Proof by contradiction , Rational and irrational numbers

Prove that if the irreducible rational fraction p/q is a root of the polynomial $P (x)$ with integer coefficients, then $P (x) = (qx – p) Q (x)$, where the polynomial $Q (x)$ also has integer coefficients.

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