Question

Topic: Discrete Math proofs

Question 1(a): is 0.103103103103..... a rational number? if so determine algebraically what rational number it is, if not, explain why not.

b) is 12 divisible by 60?

c) Let a, b, c, d ∈ ℤ. if (a-b)² is even and ab is even, what can be concluded about the parity of a and b ?

please put clear handwriting, thank you.

Answer 1

(b) clearly 60 =12*5

Thus 12 is divisible by 60

(c) Given that (a-b)^2 and ab are even for a,b in Z

Then a-b is even since its square should be a multiple of 4 since in a perfect square every prime number should

occur at least twice

Now if both of them are odd then ab is odd which is not true

If one of them is even the other is even for

a-b=2m for m in Z

a is even means b=a-2m is even

Thus both a and b are even.