CPSC 2190 M01
- (6 points) Consider the affine cipher with encryption function f(x) = (7x+5) mod 26. The decryption function is d(y) = (ay +b) mod 26, for a,b ∈ Z26. Find a and b. Recall that Zn = {0,1,2,·· ,n − 2,n − 1}.
- Consider the following statements, where a,b,m ∈ Z and m >
a ≡ b (mod m) =⇒ a ≡ b (mod 2m) |
(1) |
a ≡ b (mod 2m) =⇒ a ≡ b (mod m) |
(2) |
(a) (3 points) Is statement (1) true or false? Prove or disprove it. (b) (3 points) Is statement (2) true or false? Prove or disprove it.
- (4 points) Use Fermat’s little theorem to compute 7^{2019 }(mod 11).
- (4 points) Let f(x) = 10x^{2 }+ 2x + 4. Show that f(x) is O(x^{2}), using the definition of
O(·).
- (4 points) Let A,B be square matrices with elements in R. Prove by mathematical induction that if AB = BA then A^{n}B = BA^{n }for all n ∈ N.
- (a) (3 points) Use the Euclidean algorithm to find the gcd of 132 and 90.
- (3 points) Use your above work to find s,t ∈ Z such that gcd(132,90) = 132s+90t.
- (2 points) Did you find an inverse of 90 modulo 132? If so, what is the inverse? If not, why not?
- (a) (4 points) Use back substitution to find the inverse of 23 modulo 31 in Z31.
- (2 points) Find a solution in Z31 to the linear congruence 23x ≡ 5 (mod 31).
- (6 points) There are certain things whose number is unknown. If we count them by twos, we have one left over; by threes, we have one left over; and by fives, four are left over; by sevens, zero are left over. How many things are there? Find the smallest positive integer that satisfies this condition.
- (6 points) In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
Use mathematical induction to prove that
H_{1 }+ H_{2 }+ ··· + H_{n }= (n + 1)H_{n }− n.