Assignment 3

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 ab (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 >
 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
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