By tracing successive value of r find the output

QUESTION 1–

By tracing successive value of r find the output from the following algorithm when

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a) n =13, b) n =20

begin

Input n;

R:=n;

While r>5 do

R; = r -5;

Output r; end

What can be said about n when the output from the algorithm is zero?

Question 3-
Prove by induction on n, that
(1×2) + (2×3) + (3×4) + ….. + n (n+1) = 1/3 n (n+1)(n+2)
For all integers n>1

Question4
Consider the following subsets of the set Z of all integers.
For each of the following propositions state whether it is true or false. In each case brief reason for your answer.

Question 5-
A relation R on Z is given by x + 3y is an even integer.
a) Show that if integers x and y are R-related then either x and y are both even or x and y are both odd.
b) With the help of part (a) prove that R is an equivalence relation.