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