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Assignment 3 COMP2130 Winter 2014. Due Feb. 28, 2014 Obey all instructions given in the ROASS document. You are not allowed to use a calculator or computer on this assignment. Q1) Prove that the sum of any two odd integers is even. Is it true that the sun of any two even intgers is a multiple of 4? Prove your answer. Q2) Prove that a number is divisible by 25 only if the last two digits (treated as a two digit number) is divisible by 25. Q3) Is 123456789002 a prime? Is 444480654321 a prime? Justify your answers. Q4) How many 0’s does 21! end in. Show your work. Q5) Let m,n,a,b,d Z with d > 1. If m n(mod d) and ab (mod d), is m+a n+b (mod d). Prove it. Q6) What are the last two digits of 2999. Q7) Prove that n2+3n +5 is never divisible by 121 for any n Z+. Hint: See which of 11 cases need to be investigated further. Q8) Solve. Make sure your answer is in the modulus the question started with. a) 3x+5 12 (mod 5) b) 2x+3 -1 (mod 10) 7x2 4x 6 Q9) Simplify , where x R and x > 2 2x x 6 . Show steps. Q10) Show that (2 + 5 )/3 is irrational. Hint: see Example 4.7.1 in the book. You may assume that irrational. 5 is Q11) Prove that there are an infinite number of primes that are congruent to 3 (mod 4). Instead of using the equation Q=p12*p22*…*pn2 +2, you must use Q=4p1*p2*…*pk+3. Q12) What is the gcd(17,99)? Find an integer solution to the equation 17n +99 m = 2. What is the multiplicative inverse of 17 (modulo 99)? Use the most efficient algorithms; i.e. the Extended Euclidean Algorithm