MathDB

i.) The polynomial

x^{2 \cdot k} + 1 + (x+1)^{2 \cdot k}

is not divisible by

x^2 + x + 1.

Find the value of

k.

ii.) If

p,q

and

r

are distinct roots of

x^3 - x^2 + x - 2 = 0

the find the value of

p^3 + q^3 + r^3.

iii.) If

r

is the remainder when each of the numbers 1059, 1417 and 2312 is divided by

d,

where

d

is an integer greater than one, then find the value of

d-r.

iv.) What is the smallest positive odd integer

n

such that the product of

2^{\frac{1}{7}}, 2^{\frac{3}{7}}, \ldots, 2^{\frac{2 \cdot n + 1}{7}}

is greater than 1000?

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