Mathematics 228(Q1), Assignment 1
Due : Friday, January 19, 2007
Exercise 1.(10 marks) Assuming l,mand nare integers, show the following.
(a) If m≥nthen l+m≥l+n.
(b) If m≤nthen −n≤ −m.
Exercise 2.(10 marks) Let Xbe a non-empty subset of integers that is bounded above. Show that X
contains a greatest element.
Exercise 3.(10 marks) Prove that every odd integer is of the form 4k+1 or 4k+3 for some integer k. (Hint
: Use the Division Algorithm.)
Exericse 4.(10 marks) Verify the following statements.
(a) If c|mand c|nthen, for any intergers xand y,c|(xm +yn).
(b) Let cand dbe divisors of n. If (c, d) = 1 then cd|n.
Exercise 5.(20 marks) Calculate gcd(m, n) and express it as an integral linear combination of mand n.
(a) m= 111, n= 126.
(b) m= 89, n= 144.
(c) m= 1331, n= 3113.
(d) m= 12449, n= 30031.
Exercise 6.(10 marks) Let nbe an integer. Show (n+ 1, n2−n+ 1) divides 3.(Hint : Can you write 3 as
an integral linear combination of n+ 1 and n2−n+ 1 ?)