3.E: Exercises

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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,

when \(3^{1798}\) is divided by \(28.\)
when \(2^{1798}\) is divided by \(28.\)
when \(7^{5453}\) is divided by \(8.\)
when \(3^{135}+15^2\) is divided by \(7.\)

Exercise \(\PageIndex{7}\):

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 \(a^2 \equiv 1(\mod 8)\).