number theory
Number theory | Definition, Topics, & History | Britannica
0 is not a natural number? it depends on the mathematical community, if they design the norm(standard) that 0 i a natural number, then it is.
but in a default condition, we normally set 0 is not natural number.
b|a
Vertical bar sign in Discrete mathematics, means b divides a, b is a factor of a.
division theory ?
Euclidean division - Wikipedia
overkill 矫枉过正
Assignment9
- Express as concisely and accurately as you can the relationship between b|a and a/b.
a/b is a notion that denotes the rational number a divided by b.
b|a denotes the relation that b divides a, i.e, there is an integers q such that a=bq.
thus b|a only if a/b is an integer, b≠0, a>b, |q|>|r|
- prove your answer
(a) 0|7, False, a|b includes the requirement a≠0
(b) 9|0, true, 0=0x9, so $∃q$[0=q.9]
(c) 0|0, False, the same as (a)
(d) 1|1, true, 1=1x1, so $∃q$[1=q.1] (e) 7|44, false, ¬$∃q$[44=q.7]
(f) 7|(−42), true, $∃q$[-42= q.7]
(g) (−7)|49, true, $∃q$[49= q.-7]
(h) (−7)|(−56), true, $∃q$[-56= q.-7] (i) ($∀n$ ∈ Z)[1|n], true, for $∀n$ ∈ Z, $∃q$[n= q.1]
(j) ($∀n$ ∈ N )[n|0], true, for $∀n$ ∈ N(n≠0), $∃q$[0= q.n]
(k) ($∀n$ ∈ Z)[n|0], false, because , (n=0)∈ Z
- Prove all the parts of the theorem
(a) a|0, a|a ; true (b) a|1 if and only if a = ±1 ; prove: 1= qa = |q.a|=|q|.|a|, so |q|=|a|=1 (c) if a|b and c|d, then ac|bd (for c ≠ 0) ; true (d) if a|b and b|c, then a|c (for b ≠ 0) ; true (e) [a|b and b|a] if and only if a = ±b ; true (f) if a|b and b ≠ 0, then |a| ≤ |b| ; true