MOE H3 Math Proofs and Reasoning Problem Set 2

MOE H3 Math Mathematical Proofs and Reasoning Problem Set 2 1. Is each of the following statements true or false? Give a proof if it is true, and give a counter-example if it is false. (a) For each pair of real numbers x and y, if x + y is irrational, then x is irrational and y is irrational. (b) For each pair of real numbers x and y, if x + y is irrational, then x is irrational or y is irrational. (c) For each pair of nonzero real numbers x and y, if x is rational and y is irrational, then xy is irrational. 2. (Y2023 Q5) (a) Let p be a prime number greater than 2. Write down the possible remainders of p when divided by 4. (b) Fermat’s Little Theorem states that if p is prime and a is an integer which is not divisible by p, then ap−1 ≡ 1 (modulo p). Use Fermat’s Little Theorem to prove that if p is a prime number greater than 2, and there exists an integer z such that z2 ≡ −1 (modulo p), then p is not congruent to 3 (modulo 4). (c) Write down the possible remainders of w2 when divided by 8 where w is an integer. 3. Unique Factorization Theorem states that: Every integer n >1 has a unique standard factored form. i.e. there is exactly one way to express n = pk1 1 pk2 2 · · ·pkt t where p1 < p2 < · · ·< pt are distinct primes and k1, k2, · · ·, kt are some positive integers. Use the Unique Factorization Theorem to prove that, if a positive integer n is not a perfect square, then √n is irrational. 4. Prove or disprove the statement If a2 | b2, then a | b for all integers a, b. 5. Determine whether each of the following real numbers is rational or irrational. Justify your answers. (a) √ 3 + √ 5; (b) √ 2 + √ 8; (c) 1 + √ 2 1 + √ 3 .

H3 Math (Mathematical Proofs and Reasoning) Problem Set 2 6. Prove that there does not exists a smallest positive real number. 7. Prove that each finite decimal may be written as an infinite decimal in two distinct ways: a0.a1a2 . . . an−1an = a0.a1a2 . . . an−1anb0 = a0.a1a2 . . . an−1(an − 1)b9, where an > 0 if n >0. (Here b0 means an infinite tail of 0’s, and b9 an infinite tail of 9’s.) 8. Prove that each real number is represented by a unique infinite decimal unless it is representable by a finite decimal, in which case it is representable by precisely two infinite decimals as described in the previous problem. 9. Prove that if there are no nonzero integer solutions to the equation xn + yn = zn, then there are no nonzero rational solutions. Page 2

H3 Math (Mathematical Proofs and Reasoning) Problem Set 2 Hints 1. Hint for Question 2. Do we have all conditions necessary to apply Fermat’s Little Theorem? You may want to consider proof by contradiction or contrapositive. 2. Hint for Question 3. You may use the fact that n is a perfect square if and only if its standard factored form is n = pk1 1 pk2 2 · · ·pkt t where all the ki’s are even. 3. Hint for Question 4. Use prime factorization. 4. Hint for Question 5. May use the fact that if a positive integer n is not a perfect square, then √n is irrational. 5. Hint for Question 8. Requires induction. Page 3