a. Use proof by contradiction to show that for any integer

QUESTION:

a. Use proof by contradiction to show that for any integer n, it is impossible for n to equal both \(3q_{1}+r_{1}\) and \(3q_{2}+r_{2}\), where \(q_{1}, q_{2}, r_{1}\), are integers, 0 \(\leq r_{1}\) < 3, 0 \(\leq r_{2}\) < 3, and \(r_{1} \neq r_{2}\).

b. Use proof by contradiction, the quotient-remainder theorem, division into cases, and the result of part (a) to prove that for all integers n, if \(n^{2}\) is divisible by 3 then n is divisible by 3.

c. Prove that \(\sqrt{3}\) is irrational.

Text Transcription:

3q_1 + r_1

3q_2 + r_2

q_1, q_2, r_1

leq r_1

leq r_2

r_1 neq r_2

n^2

sqrt 3