7/22/2019 MA14 U16
1/8
n! n
ba
b a
a > b
b
a
= 0
a N b, c N
a
b
c
= a
bc
.
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q1 =a
b
e q2=
ab
c
.
a= bq1+r1, com r1 b 1
ab
= q1 = cq2+r2, com r2 c 1.
a= bq1+r1= b(cq2+r2) +r1= bcq2+br2+r1.
br2+r1 b(c 1) +b 1 =bc 1,
q2 a bc
q2= a
bc
.
c a b
a b c
p m
Ep(m) p m
p m
Ep(n!)
p
n!
7/22/2019 MA14 U16
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a m a >1
n
an
> m
A= {an; n N}
an m n N A
r N x ar x A a 2
ar+1 2ar > ar,
ar+1 A ar A
n p
Ep(n!) =
n
p
+
n
p2
+
n
p3
+
r pi > n i r n
pi
= 0 i r
n
n= 0
m m < n p n
p, 2p, . . . ,
n
p
p.
Ep(n!) =
n
p
+Ep
n
p
!
.
Ep
n
p
!
=
n
p
p
+
n
p
p2
+
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Ep(n!)
n= pq1+r1
q1 = pq2+r2
. . .
qs1= pqs+rs
q1 > q2 > s qs < p
Ep(n!) =q1+q2+ +qs.
10!
Ep(10!)
p 10 E2(10!) = 5 + 2 + 1 = 8, E3(10!) = 3 + 1 = 4, E5(10!) = 2
E7(10!) = 1
10! = 2834527.
10! 10!
a1, . . . , am, b b= 0 a1+ +am
b
a1b
+ +
amb
.
qi ri
ai b i = 1, . . . , m
ai= bqi+ri
a1+ +am= (q1+ +qm)b+r1+ +rm.
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a1+ + am b
q1+ +qm
r1+ +rm
b 1
a1, . . . , am, b b = 0
(a1+ +am)!
a1! am! .
p
i
a1+ +am
pi
a1
pi + +
am
pi .
Ep((a1+ +an)!) Ep(a1!) + +Ep(am!),
Ep(n!) p n
p
p, n N p
n= nrpr +nr1pr
1 + +n1p+n0
p n
Ep(n!) =n (n0+n1+ +nr)
p 1 .
0 ni< p n
p
= nrp
r1 + nr1pr2 + + n2p + n1
n
p2
= nrpr2 + nr1pr3 + + n2
. . .n
pr
= nr
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Ep(n!) =n
p
+ n
p2
+ + n
pr
=
nrpr 1
p 1 +nr1
pr2 1
p 1 + +n1 =
nrpr +nr1p
r1 + +n1p+n0 (nr+nr1+ +n1+n0)
p 1 =
n (n0+n1+ +nr)
p 1 .
100!
2 5 10000!
10000!
10000!
n 5
n! 584
n 37
n!
a1, . . . , am N b N
a1b
+ +
amb
a1+ +am
b
a1b
+ +
amb
+m.
m, n N (m, n) = 1
(m+n 1)!m!n!
N.
m,n,b N b= 0
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2m
b
+
2n
b
m
b + n
b +
m+n
b
(2m)!(2n)!
m!n!(m+n)!
n, m N (n m)! [(n!)m, (m!)n]
(n!)(n1)! (n!)!
n, a1, . . . ar N d = (a1, . . . , ar)
d(n 1)!
a1! ar!.
7/22/2019 MA14 U16
8/8
1 +1
4+
1
9+
1
16+
1
25+
2/6