Mathematics 220
1. Prove that
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2. Prove that
√
Homework 6
Due Feb. 24/25
6 is irrational.
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2 + 3 is irrational.
3. Prove that if the integers a and b are both odd, then a2 + b2 cannot be a perfect square.
4. Prove that the number 123456782 cannot be represented as a2 + 3b2 for any integers a
and b. (Hint: Consider the remainder mod 3).
5. (a) Prove that there are infinitely many primes p such that p ≡ 3 mod 4. Hint: try to
proceed the same way as in Euclid’s proof of the statement that there are infinitely
many prime numbers; but instead of making the number N = p1 , . . . pn + 1, make
a number N that is definitely congruent to 3 modulo 4 (and that still differs by 1
from a number that is divisible by all of p1 , . . . pk ).
(b) Could this proof have worked for the primes congruent to 1 modulo 4?
6. Find the last digit of the number 20162016 .
7. Are the following statements True or False (provide complete proofs):
(a) If x and y are both irrational, then x + y is irrational.
(b) If x and y are both irrational, then xy is irrational.
(c) If x is rational and y is irrational, then xy is irrational.
(d) If x 6= 0 is rational and y is irrational, then xy is irrational.
√
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(e) If a, b ∈ Q and ab 6= 0, then a 3 + b 2 is irrational.
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