Let a and b be two integers...?

Let a and b be two integers and n be a natural number. Show that n divides a − b if and

only a and b have the same remainder when divided by n.

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  • 2 years ago
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    If a = nc + k and b=nd + k (where k, c, and d are integers), then a - b = (nc + k) - (nd + k) = n(c - d), which, since c - d is an integer, means n divides a - b.

    If n divides (a - b), then a - b = nc for some integer c, and c = a/n - b/n

    a/n = {some integer d} + k/n, and b/n = {some integer e} + j/n, where k and j are < n.

    This makes c = {an integer} - {an integer} + k/n - j/n.

    Since c is an integer (k/n - j/n) must also be an integer. Since j and k are < n, it must be that k/n = j/n.

    This means a and b have the same remainder when divided by n.

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