need help!!! plz write clearly # 2. Let a and b be non-zero coprime integers. Show...
(a) If a | bc, show that a | b*gcd(a,c). (b) If a, b are coprime integers and c | at and c | bt, show that c | t. (c) If a, b, c are integers with a, c coprime, prove that gcd(ab, c) = gcd(b, c).
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
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...
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
Prove all non-zero integers a and b, if gcd(a, b) = d then for all non-zero integers x if a|x and b|x then ab|dx.
(d)n- 1013 2. Let a, b, c, d be integers. Prove the statement or give a counterexample (a) If (ab) c, then a |c and alc. (b) If a l b and c|d, then ac bod (c) If aYb and alc, then aYbc. (d) If a31b4, then alb. (e) If ged(a, b) 1 and alc and b c, then (ab) c. Here a and b are relatively prime integers, also called coprime integers.] rherF and r is an integer with...
(11) Let A-{2" . 3", | n and m are non-zero integers). Show that 1 єА. (11) Let A-{2" . 3", | n and m are non-zero integers). Show that 1 єА.
Let a and b be non-zero elements of a principal ideal domain R, and let 1 = (a) and I = (6). Show that the following are cquivalent: (i) I and I are comaximal. (ii) In J = II. (iii) ab is a least common multiple of a and b. (iv) 1 = ged(a,b).
Question 2. Let a, b, c be natural numbers. (a) Suppose that g specific a, b, c eN where d > g. ged(a, b) ged(b, c). Let d ged(a, c); prove that d > g. Provide an example of (b) Let d gcd(a, b). By definition of the ged being a divisor of a, b, this implies that we may write a and b jd for some j,k E N. Prove that ged(j, k) = 1. kd -