Nombres premiers et PPCM

N
n
•
•
•
•
n
•n
•n p p ≤√n
•n
•n
p p N
p p d
d p p n d n
p
•p n q n =pq
p≤q p
p2≤pq p2≤n p ≤√n
27−1
•
•27−1 = 127 √127 ≈11,3

•p1, p2, ..., pk.
•a=p1×p2×... ×pk+ 1
pi{p1, p2, ..., pk}
pia p1×p2×... ×pkpi
•
n n
•n≥2n p1
n=p1n11≤n1< n p1>1.
n1= 1 n=p1n
n1≥2n1p2
n=p1p2n21≤n2< n1< n p2>1.
n2= 1 n=p1p2n
n2≥2ni>1
ni
nk
n p1, p2, ..., pk
•24 = 23×3
•150 = 2 ×3×52.
n
n
a b
b a b
a b

•b a a =bq a b q
•b a
b a =pα1
1×pα2
2×...×pαk
k
b=pβ1
1×pβ2
2×... ×pβr
rr≤k i r αi≥βi
a=pα1−β1
1×pα2−β2
2×...pαr−βr
r×pαr+1
r+1 ×... ×pαk
k×pβ1
1×pβ2
2×... ×pβr
rb a
272 = 24×17
a b
a b a b a
b
•N
|ab|a b
•
•
•
b b
12 = 22×3 60 = 22×3×5
b b = 2α×3β×5γ0≤α≤2 0 ≤β≤1 0 ≤γ≤1
b b γ = 1
b
•m=a b m a b m a
b
•M a b M m M =mq +r
0≤r < m qZ
a b M m r =M−mq m a b 0≤r < m
r= 0
m M

Z212x= 16y
z= 12x= 16y z
= 48 = 12 ×4 = 16 ×3z= 12 ×4×n= 16 ×3×n nZ
x= 4n y = 3n nZ
a b k
ka kb =|k|a b
•(a, b) (|a|,|b|)a, bN
•m=a b km ka kb k a b ≥ka
kb
•M=ka kb M =λka =λ0kb λ, λ0Na b M
k
M
k
a b m ka kb ≥k a
b
•ka kb =|k|a b
a b
a b a b
a b
54 = 2 ×3384 = 22×3×7 2 ×3 = 6
a b
a b a
b a b
54 = 2 ×3384 = 22×3×7 22×33×7 = 756
a b (a, b)×(a, b)ab
×6×756 = 4536 = 54 ×84