Q:

Discrete Matha) Find the quotient q and the remainder r as defined in the Division Algorithm so that a = qb + r where a = βˆ’65 and b = 11.b) Find the gcd(1200, 560). Show some of your computations.c) Prove that if b|a and b|c then b|(a + c).

Accepted Solution

A:
Answer:Part c: Contained within the explanationPart b: gcd(1200,560)=80Part a: q=-6 Β  Β  Β  Β  r=1Step-by-step explanation:I will start with c and work my way up:Part c:Proof:We want to shoe that bL=a+c for some integer L given:bM=a for some integer M and bK=c for some integer K.If a=bM and c=bK,then a+c=bM+bK.a+c=bM+bKa+c=b(M+K) by factoring using distributive propertyNow we have what we wanted to prove since integers are closed under addition. Β M+K is an integer since M and K are integers.So L=M+K in bL=a+c.We have shown b|(a+c) given b|a and b|c.//Part b:We are going to use Euclidean's Algorithm.Start with bigger number and see how much smaller number goes into it:1200=2(560)+80560=80(7)This implies the remainder before the remainder is 0 is the greatest common factor of 1200 and 560. So the greatest common factor of 1200 and 560 is 80.Part a:Find q and r such that:-65=q(11)+r We want to find q and r such that they satisfy the division algorithm.r is suppose to be a positive integer less than 11.So q=-6 gives:-65=(-6)(11)+r-65=-66+rSo r=1 since r=-65+66.So q=-6 while r=1.