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Mathematics LibreTexts

3.E: Exercises

  • Page ID
    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 \(8.\)
    4. when \(3^{135}+15^2\) is divided by \(7.\)

    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 \(n^3-n.\)

    Exercise \(\PageIndex{9}\)

    Compute the last two digits of \(9^{1600}\).

     

    Exercise \(\PageIndex{10}\)

    Show that \(a^2+b^2  \not\equiv  3(\mod 4)\) for any integers \(a\) and \(b\).

    Exercise \(\PageIndex{11}\)

    Let \(a\) be an odd integer. Show that \(a^2 \equiv 1(\mod 8)\).