Exercise 1: Exactly Where the Contradiction Arises — Possible Solution
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The proof begins by assuming √2 = a/b, where a/b is written in LOWEST
TERMS - meaning a and b share no common factor other than 1. This
"lowest terms" assumption is the specific claim the contradiction will
end up violating.

Working through the algebra (squaring both sides, then reasoning about
even and odd numbers) forces two separate real conclusions: first,
that a itself must be even, and then, substituting a = 2k back in,
that b must also be even. Both of these conclusions follow directly
and necessarily from the original assumption - there's no error or gap
in the reasoning up to this point.

The contradiction arises the moment you notice what "a is even AND b
is even" actually means: if both a and b are divisible by 2, they
share a common factor of 2. But the very first step of the proof
assumed a/b was already in lowest terms, meaning a and b were
SUPPOSED to share no common factor at all. The proof has derived a
direct violation of its own starting assumption, using nothing but
valid algebraic and logical steps along the way.

Since the assumption ("√2 can be written as some fraction in lowest
terms") led, through entirely valid reasoning, to something that
directly contradicts itself, the only way to resolve the contradiction
is to conclude the original assumption must have been false in the
first place - meaning no such fraction in lowest terms can actually
exist, which is exactly what "√2 is irrational" means.

ANSWER: The contradiction arises because the proof assumes a/b is in
lowest terms (no shared factor), but valid reasoning from that
assumption forces both a and b to be even - meaning they DO share a
factor of 2, directly violating the starting assumption. Since the
assumption led to its own contradiction through valid steps, the
assumption itself must be false, meaning √2 cannot be written as any
fraction at all.

WHY THIS WORKS AS AN ANSWER
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This traces the specific assumption being violated (lowest terms) and
shows precisely how the derived conclusion (both even) directly
contradicts it, rather than vaguely gesturing at "a contradiction
happens somewhere."