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2 December, 08:23

Discrete Math

a) Find the quotient q and the remainder r as defined in the Division Algorithm so that a = qb + r where a = - 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).

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  1. 2 December, 12:20
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    Part c: Contained within the explanation

    Part b: gcd (1200,560) = 80

    Part a: q=-6 r=1

    Step-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+bK

    a+c=b (M+K) by factoring using distributive property

    Now we have what we wanted to prove since integers are closed under addition. 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) + 80

    560=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+r

    So r=1 since r=-65+66.

    So q=-6 while r=1.
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