Discrete mathematics proof

(See attached file for full problem description) --- Let d,m and n be positive integers with m>1 and m≡ 1 (mod d), let n= c0+mc1+m2c2+m3c3+…+mrcr be the base=m expansion of n, and let f = c0+c1+c2+c3+…+cr Prove that n is divisible by d if and only if f is divisible by d. ---

RSA encryptions

(See attached file for full problem description) --- Consider the RSA encryption system given by p=43,q=59, and e=13 i) Find d such that ed ≡ 1 (mod (p-1)(q-1)) ii) Decode the message : 1552 2069 1178 1637 1975 Using the convention A = 00, B = 01, …, Z = 25 ---

Discrete mathematics proofs

1) Prove that all integers a,b,p, with p>0 and q>0 that ((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q Or give a counterexample 2) prove for all integers a,b,p,q with p>0 and q>0 that ((a-b)mod p) mod q=0 if and only if (a mod p) mod q = (b mod p) mod q Or give a counterexample. 3) let p and q be positive integers with 0 < p < q and gcd(p,q) = 1 and let a and b be integers with 0<=a <=p-1 and 0<=b<=p-1 Prove that there exists an integer x such that (x mod p) mod q = a and (x mod q) mod p = b

Extended Euclidian Algorithm Proofs

(See attached file for full problem description) --- Given positive integers a and b, the extended Euclidian algorithm constructs sequences qn, rn, sn and tn, which are defined recursively as follows: q0=0, q1=0, qn= q└ rn-2/ rn-1 ┘ for n>=2; r0=a, r1=b, rn= rn-2 - qnrn-1 for n>=2; s0=1, s1=0, sn= sn-2 - qnsn-1 for n>=2; t0=0, t1=1, tn= tn-2 - qntn-1 for n>=2; the sequences terminating if rn=0 . Prove the following i) The extended Euclidian algorithm always terminates, i.e., there exists n>=2 such that rn=0 (hint : first show that if k>0 and rk>0 then 0<= rk+1click for more

Hasse Diagrams

(a) List the ordered pairs that belong to the relation. My ANS: (a,a),(b,a)(b,b),(c,a),(c,b),(c,c),(d,a),(d,b)(d,d) (b) Find the (boolean) matrix of the relation. < my answer file attached as Mr.jpg> I did review another answer posted along the same question, but uses a different shape. I think my matrix is correct based on my pairs, but i'm concerned about the pairs.