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Mathematics LibreTexts

3.E Exercises

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    7526

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    Exercise \(\PageIndex{1}\):

    Let \(a, b, c \in \mathbb{Z} \), such that \( a \equiv b (mod\,n). \)

    Show that  \(ac=bc(mod\,n). \)

    Exercise \(\PageIndex{2}\):

    Find the remainder when \((201)(203)(205)(207)\)  is divided by \(13.\)

    Exercise \(\PageIndex{3}\):

    Show that the sum of 2 odd integers is even.

    Exercise \(\PageIndex{4}\):

    Given that February 14, 2018, is a Wednesday, what day of the week will February 14, 2090 be? 

    Exercise \(\PageIndex{5}\):

    Find the remainder when 81789  is divided by 28.

    Exercise \(\PageIndex{6}\):

    Find the remainder,

    1.  When \(3^{1798}\) is divided by \(28.\)
    2.  When \(2^{1798}\) is divided by \(28.\)
    3.  When \(7^{5453}\) is divided by \(8.\)

    Example \(\PageIndex{7}\):

    Given a positive integer \(x,\) rearrange its digits to form another integer \(y.\) Explain why \(x-y\) is divisible by \(9.\)

    Exercise \(\PageIndex{8}\)

    Prove that for all integer \(n\geq 1,\,6\) divides \(n^3-n.\)

    Answer

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