James Chen TMUA Paper 2 · Question 15

A student attempts to prove, by contradiction, that $\sqrt[n]{2}$ is irrational for all integers $n>2$ . Their proof begins as follows:

\text{Assume }\sqrt[n]{2}=\frac rs,\text{ where }r,s\text{ are integers.}

2=\frac{r^n}{s^n}

The student despises the idea of infinite descent due to personal trauma or whatever, so shall not use it in their proof. Which of the following theorems is most useful to the student in completing their proof?

AThis option knows the right theorem, but is too small to contain it

1^n+2^n+\ldots+a^n

is a polynomial in

a

for integers

a

and

n>2

a^n\equiv a\pmod b

when n is an integer greater than

2

a^k+b^k=c^k

has no integer solutions with

k>2

examday

James Chen TMUA Paper 2 · Q15