Rudiments d`arithmétique dans N

N
(a, b)∈N2b a a k ∈N
a=kb b |a a b b a
1 2 3 6 6 1
0
(a, b, c)∈N3
a|a
a|b b |a a =b
a|b b |c a |c
|N
≤R
3 12 3 12 9
12
22 4
22 = 4 ×4+6
22 = 4 ×5+2

4 6
22 4
(a, b)∈N×N∗(q, r)∈N2
a=bq +ret 0 ≤r < b.
q r
a b
666999 123
a b
(a, b, c, k)∈N4a=bk +cD(a, b) = D(b, c)D(p, q)
p q
a b d
a b
d|a d |b
∀δ∈N(δ|aet δ|b) =⇒δ|d
PGCD(a, b)a∧b
a b a b
≤ |

(a, b)∈N2k∈N
PGCD (ka, kb) = k·PGCD(a, b).
d= PGCD(a, b)
a0b0a=da0b=db0a0b0
1
a, b, c
a|bc PGCD(a, b)=1 a|c
PGCD(a, b)=1 a b
a b m
a b
a|m b |m
∀µ∈N(a|µet b|µ) =⇒m|µ
PPCM(a, b)a∨b
a, b ∈N
PGCD(a, b)·PPCM(a, b) = ab.
p∈N\ {0,1}1p
2 3 5 7 11 13

p a, b p ab a b
2
n2n
√n
100
10
2 3 5 7 100
π(n)n
π(n)∼n
ln(n).

n≥2m≥1p1, p2,...pm
v1,...vm
n=
m
Y
k=1
pvk
k.
n≥2n n =
m
Q
k=1
pvk
k
n q
q=
m
Y
k=1
puk
k∀k∈J1, mK0≤uk≤vk.