Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On...
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?
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...
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...
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If m and n are any positive integers, again relatively prime
number thoery just need 2 answered 2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
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
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
For any positive integer n, Euler’s totient or phi function, Φ(n), is the number of positive integers less than n that are relatively prime to n.? What is Φ(55) ?
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.
Show that every positive integer n, there is a string of n consecutive integers where first integer is even, the second is divisible by a perfect square(other than 1), the third by a perfect cube(other than 1), etc..., and the nth is divisible by the nth power of an integer(other than 1). Then find an example for n = 3.