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Mathematics LibreTexts

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,

  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 \(9.\)