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Divisibility:
Basic properties:
(a, b, c, m, x, y are all integers)
| (1) | a | b implies a | bc for any integer c |
| (2) | a | b and b | c imply a | c |
| (3) | a | c and a | c imply a | (bx+cy) for any integers x and y |
| (4) | a | b and b | a imply a = b or -b |
| (5) | a | b,a>0,b>0, imply a <= b |
| (6) | if m=/0, a | b implies and is implied by ma | mb |
These divisiblity properties are very easy to prove.
Division algorithm Given any integers a and b, with a > 0, there exist unique integers q and r such that b = qa + r, where 0 £ r < a.
Again, this is pretty obvious. If we divide an integer a by another integer b, we will get a quotient, q, and a remainder, r.