PMATH340 LN8

Also we have ac qn rq n r qq n2 rq r q

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: e remainder when divided by n, say r￿ . So there exist q, s, q ￿ , s￿ ∈ Z so that a = qn + r b = sn + r c = q￿ n + r￿ d = s￿ n + r￿ . Using this we have that a + c = qn + r + q ￿ n + r￿ = (q + q ￿ )n + (r + r￿ ) ≡ (r + r￿ )(mod n) ≡ (s + s￿ )n + (r + r￿ )(mod n) ≡ (sn + r) + (s￿ n + r￿ )(mod n), so that a + c ≡ b + d(mod n). Also we have ac = (qn + r)(q ￿ n + r￿ ) = qq ￿ n2 + (rq ￿ + r￿ q )n + rr￿ ≡ rr￿ (mod n) ≡ ss￿ n2 + (rs￿ + r￿ s)n + rr￿ (mod n) ≡ (sn + r)(s￿ n + r￿ )(mod n) ≡ bd(mod n) so that a...
View Full Document

This note was uploaded on 02/09/2013 for the course PMATH 340 taught by Professor W.alabama during the Spring '09 term at Waterloo.

Ask a homework question - tutors are online