1. cos(n+1) = 2cos(n)cos(1) - cos(n-1)
Need help with part b) from the first question and the other 2 questions.
if b is an integer, it must be a factor of +- itself, b or -b. If it is not a factor of any natural number, it is an integer that is not a natural number and is thus 0.Question 2 was
Prove that if b is an integer and b is not a factor of k for every k (Natural Numbers), then b=0
a) is just rational root theorem. Oh you want by contradicton. AssumeMore questions (Contradiction) :
Explain why x^3+x+1=0 has no rational roots
Prove that there exists no integer a and b for which 21a+30b=1
Qeru, just a quick reminder that you needa) is just rational root theorem. Oh you want by contradicton. Assume(where a and b are non zero integers) is a root. Then:
. The parity of the LHS is different to the parity of the RHS so contradiction.
How does this follow? You could have some combination of a and b which equals zero (since either a or b could be negative).Further, examiners will not necessarily know what you mean by LHS and RHS have different parity. It would be simpler in this case to noteand so
, producing the necessary contradiction.
True... an alternative is then thatHow does this follow? You could have some combination of a and b which equals zero (since either a or b could be negative).
How could we do Question 4)1. Prove that there is no integer that leaves a remainder of 2 on division by 6, and also leaves a remainder of 7 on division by 9.
2. Prove thatfor any value
3. Prove that no integer has a square that leaves a remainder of 3 on division by 4.
4. Prove that if a right angled triangle has sides of lengths,
, and
(where
is the length of the hypoteneuse), and another right angled triangle has sides of lengths
,
, and
(where
is the length of the hypoteneuse), then at least one of
,
, and
is not an integer.
5. Prove thatis irrational.
6(a). Suppose thatand
are positive integers. Find real constants
and
such that
(b). Hence, or otherwise, show that, ifis a positive integer, then
(c). You may use, without proof, the fact that
in order to prove thatis irrational.
Assume a,b and c are integers. We know from pythagoras:How could we do Question 4)