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ID: 3905076 • Letter: A
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Question 2 (24 points) Prove the following statements. a) Let x, y E Z. Then ry is odd if and only if x is odd and y s odd. b) Let r, y ? R. Then (z + y)2-2,2 + ?? if and only if r-0 or y-0. c) Let a,DE N. Then a-: gcd(a, b) if and only if alb. and only if x- U or-0.Explanation / Answer
a) xy is odd if and only if x is odd and y is odd
if x is odd and it is multiplied by even , then the product is even as any number multiplied with even will give even result so y has to be odd if xy has to odd. This means that x and y have to be odd to have xy as odd. Other way if xy odd , then x and y
has to be odd because product will be even if any one of x or y is even and hence the
other way also x and y are odd
b) (x + y)^2 = x^2 + y^2 if and only if x = 0 or y = 0
(x + y)^2 = x^2 + y^2 + 2xy (well known formula)
so if we have
x^2 + y^2 + 2xy = x^2 + y^2
2xy = 0
xy = 0 This means either x = 0 or y = 0
otherway round if we have either x = 0 or y = 0 then
(x + y)^2 = x^2 + y^2 + 2xy = x^2 + y^2 (as xy = o if we have either x = 0 or y=0)
Hence Proved both ways
c) a = gcd(a,b) if a|b
gcd(a,b) stands for greatest common divisor which divides a nd b. so if a divides b then a is the greatest common divisor for a and b because we can not a number more
than a as a divisor for a. So if a divides b then it has to be the greatest common
divisor for a and b
common divisor which divides a and b
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