Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
27. (a) Let m and n be integers > 1 which are relatively prime. Show that the map f : Z → Z/mZ × Z/nZ whith f(x) = (x + mZ, x + nZ) is surjective (b) Prove the Chinese Remainder Theorem: If m and n are relatively prime integers > 1 and if a and b are any integers, then there exists a E Z such that b(mod n). a(mod m) and a a Hint: (a)] 27. (a) Let...
Suppose that d = s and and positive integers m and n (a) Show that m/d and n/d are relatively prime ged(m, n) sm +tn for some integers (b) Show that if d = s'm + t'n for s', t' e Z, then s' = s kn/d for some k e Z.
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
Suppose a and b are numbers that are relatively prime to p. Show that at least one of the three numbers, a, b or ab, must be a quadratic residue.
c and 21 Let a, b, c E Z, where a and b are relatively prime nonzero integers. Prove that if a blc, then abc.
9. Integers m, n with god(m, n) = 1 are called "relatively prime" or "co-prime". Assume now m and are indeed co-prime. (i) Show that ged(m + n,m-n) 2m and ged(m + n. m -n 2n (ii) Use part (i) to show that there are only two possible values that ged(m + n. m - n) can attain, namely 1 or 2
Problem 6: There are three users with pairwise relatively prime moduli n, n and n3. Suppose that their encryption exponents are all e3. The same message m is sent to each of them and you intercept the three ciphertexts ci mrs (mod n.), for i-1, 2, 3. (a) Show that 0 m3< nin2n (b) Show how to use the CRT to find m3 (as an exact integer, not only as m3 (mod ninns)) and, therefore also m c) Suppose that...
Let m be a positive integer and let a and b be integers relatively prime to m with (ord m a , ord m b) )=1. Prove that ord m (ab)= (ord m a) (ord m b) (Hint: Let k=ord m(a),l=ord m(b), and n=ord m(ab). Then 1≡(ab)^kn≡b^kn mod m. What does this imply about l in relation to kn?
(For this question, do not use prime factorization) Suppose that a, b and d are positive integers with d ab. Prove that there exists positive integers e and f such that ea, f b and d= ef. Further show that the values of e and f are unique if (a, b) = 1.