Proofs involving congruences

• Read carefully pages 103 through 105 (section 4.2)
• Some key questions to answer (try these without looking at the book, but after you’ve read the book):
  1. For integers \(a\), \(b\), \(n\), what is the definition of the phrase “\(a\) is congruent to \(b\) modulo \(n\)”?
  2. True or False: Every integer is congruent to \(0\), \(1\), \(2\) or \(3\) modulo \(4\) (an exactly one of these is correct).
  3. For integers \(a\), \(b\), \(k\), \(n\), prove that if \(a\) is congruent to \(b\) modulo \(n\) then \(ka\) is congruent to \(kb\) modulo \(n\).
  4. For integers \(a\), \(b\), \(c\), \(d\), \(n\), prove that if \(a\) is congruent to \(b\) modulo \(n\) and \(c\) is congruent to \(d\) modulo \(n\), then \(a+c\) is congruent to \(b+d\) modulo \(n\) and also \(ac\) is congruent to \(bd\) modulo \(n\).
  5. Show that for an integer \(n\), if \(n^2\) is not congruent to \(n\) modulo \(3\), then \(n\) is not congruent to \(0\) modulo \(3\) or to \(1\) modulo \(3\).
• Practice problems from section 4.2 (page 114): 4.15, 4.17, 4.18, 4.21