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

.

7/22/2019 MA14 U16

2/8

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

3/8

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

+

7/22/2019 MA14 U16

4/8

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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5/8

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

7/22/2019 MA14 U16

6/8

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

7/22/2019 MA14 U16

7/8

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