Question 2. Let a, b, c be natural numbers. (a) Suppose that g specific a, b,...
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)
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...
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....
T'he goal of this problem is to establish the following remarkable result: Bezout's theorern. If a, be Z50, then 3x, y є Z such that gcd(a, b) = ax + by. Here ged(a, b) denotes the greatest common divisor of a and b (i.e. the largest positive integer that divides both a and b). Throughout this problem, we'll use the notation (a) Write down five numbers that live in 2Z +3Z. What's a simpler name for the set 2Z +3Z?...
10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime. han C: oadl Prove that if p, q, and r are three different prime numbers, then p2 + q2 #r2 11.
13 a. Let Let (and new be a sequence of reel numbers and let o cael. Assume that for some NEN I calanl In), N. Prove that linn an 1 anti b. het (annsyl be the sequence defined by anti & Satan (i) Prove that to EN Lan 1.2. (11) Prove that and give its limit (an) converges C. Using the canchy's definition of continuity , prove the funetion g(x) = 2x+1 x-4 is continuous at l.
5.42 Let G be a connected graph of diameter d 2 2 and let k be an integer with 2SkSd. Prove that if Gk is a distance-labeled graph and e is an edge labeled j where 1 < j-k, then e is adjacent to an edge labeled i for every integer i with 1 i < j.
8. Let f:D → R and let c be an accumulation point of D. Suppose that lim - cf(x) > 1. Prove that there exists a deleted neighborhood U of c such that f(x) > 1 for all 3 € Un D.
3. Let M be a manifold and let G C Homeo(M) be a group acting on M. Suppose that this group action is properly discontinuous and free prove that the quotient space M/G is a manifold. For this problem properly discontinuous means that if K c M is compact then the set {ge G | g(K) n/Kメ0) is finite) and free means the only element of g that fixes any point of M is the identity. 3. Let M be...