Number Theory-Divisibility (1)

Proper divisor. We say that a is a proper divisor of b if a | b and a < b.

For example, 3 is a proper divisor of 6, but 6 is not a proper divisor of 6. Note that for any b, all divisors of b are proper except for b itself.

Nontrivial divisor. We say that a is a nontrivial divisor of b if a | b and 1 < a < b.

For example, the nontrivial divisors of 6 are 2 and 3. Note that 1 is a proper divisor of every large integer, but 1 has no trivial divisors. The following theorem states some properties of divisibility.

Theorem 2.1
Properties of Divisibility
D1: 1 | a and a | a for all a.
D2: If a | b, then a =< b
D3: If a | b and b | a, then a = b.
D4: If a | b and b | c, then a | c.
D5: For all c, a | b if and only if ac | bc.
D6: If a | b and c | d, then ac | bd
D7: If a | b and a | c, then a | (bx + cy) for all x, y

Proof of D2: If a | b, then by proper division, b = ka for some k >= 1, so ka >=a, that is b >=a.
QED

Proof D4: If a | b and b | c, then b = ka and c = jb for some k and j. Therefore c = k(ka) = (jk)a, so a | c.
QED

Proof of D7: If a |b and a | c, then b = ka and c = ja for some j, k. Therefore, bx + cy = (ka)x + (ja)y =
(kx + jy)a, so a | (bx + cy).
QED

Remarks
In order to prove that m = n, it is sometimes convenient to use D3, that is, to prove that m | n and n | m.