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22 August, 16:22

Prove Forall a, b, c elementof Z^+, if a| (b + c) and a|c then a|b.

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  1. 22 August, 20:03
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    Answer with explanation:

    If A is a positive Integer, then if A divides B, then in terms of equation it can be written as

    →B=A m, where m is any integer.

    ⇒Now, it is given that, three elements, a, b and c belong to set of Integers.

    a divides b+c, and a divides c

    then we have to prove that, a divides b.

    Proof

    →b+c = k a, where k is an integer, as b+c is divisible by a.

    →Also, c = m a, where m is an integer. Because c is divisible by a.

    →b + m a = k a

    →b=k a - ma

    →b=a (k - m)

    Since, k and m are both integers. So, k-m will be also an integer.

    Let, k-m = p

    →b=a p

    which shows that, b is divisible by a.

    Hence proved.
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