3as-sm-math1-l08
DESCRIPTION
importanatTRANSCRIPT
8
)*(
-1 . -2 :( .( .
-4
P PCM PGCD
.
-5 . -6 . -7 .
:
: . .
:
1
:
.
200m2
150m2
250m2
200m2
300m2
m2
. 250
S
6000m2 < S < 5000m2
.
:
S 150[ 200] S 200[ 250]
S 250[300]
S + 50 0[ 200] : S + 50 0[ 250] S + 50 0[300] S + 50 S + 50
300 250 200
300 250 200
m
.
.
PPCM ( 200250300 ) = 10 PPCM ( 202530)= 10 5PPCM ( 456)= 10 5 2 5 6 = 3000 : :
m = 3000
S + 50 : 3000
.
S + 50 = 3000 k , k *
S = 3000k - 50 :
:
k *
5000 < S < 6000 : 5000 < 3000 k - 50 < 6000 :
:
5050 < 3000 k < 6050
50506050 1
q1
.
d1 n = d1 .q 1 :
.
q1 :
n = d1d 2 q2
q2
.
q2
d2 q1 = d 2 .q 2 :
.
.
n n = d1 d2 ... d p : d < d2 < ...< d p :
. 1
:
n = k1 k2 ... ks
:
k1 < k2 < ...< ks
(1)...d1 d 2 ... d p = k1 k2 ... ks : d1
k1 k2 ... ks
k1 k1 d1 :
k1 d1 d 2 ... d p
k1
d1 d1 k1 : d1 = k1 :
d 2 d3 ... d p = k2 k3 ... ks
:
d 2 = k2 :
d 3 = k3
d s = ks
)
s< p
(
:
k1 k2 ... ks d s+1 ... d p = k1 k2 ... ks d s+1 ... d p = 1 :
.
:
n
p
n n = a1p a2p ... aqq :
a
a a
1 , 2,..., q :
1
:
p1, p2,...,pq
2
.
7560 :
:
7560 = 23 33 5 7
2
:
1
7560 23780 21890 2945 3315 3105 335 57 71 80000 :
4
80000 = 8 10 4 = 23 ( 2 5 ) = 23 24 54
:
80000 = 27 54
:
(8 :
5
:
:P
n = a1P a 2P ... a qq2
1
(1 + p1 )(1 + p2 ) ... (1 + pq -1)(1+ pq)
:
: :
n = a1P a 2P ... a qq
P
d N
:
1
2
0 a1 p1
:
:
a
d = a1a a 2a ... a qq2
...
0 a 2 p2
(1+ p1) (1+ p2)
a1
1
0 a q pq
.
a2
.
M (1+ pq)
d
aq
.
(1 + p1 )(1 + p2) ... (1+ pq) :
:
180
. .
::
180 = 18 10 = 32 2 10 = 22 32 5
180
:
(1 + 2 )(1 + 2 )(1 + 1)= 18
.
* 180 :
: 180
0a 2
a
0b 2
b
b =0a =0
b =1b =2b =0
a =1
b =1b =2b =0
a =2
b =1b =2
g
2a 3b 5g
0 g 1 180
g =0
1
g =1
5
0 =g
3
g =1
15
0 =g
9
g =1
45
0 =g
2
g =1
10
0 =g
6
g =1
30
0 =g
18
g =1
90
0 =g
4
g =1
20
0 =g
12
g =1
60
0 =g
36
g =1
180
(9 : :
n1,n2,...,np
n1,n2,...,np :
.
:
PGCD(1000480250)
:3
1000 = 103 = ( 2 5 ) = 23 53480 = 48 10 = 3 16 2 5 = 25 3 5 :
250 = 25 10 = 52 2 5 = 2 53PGCD (1000480250 )= 2 5 =10
(10 : :
n1,n2,...,np n1,n2,...,np :
. :
PPCM(1000480250) :
1000 = 2 3 53 480 = 25 3 5250 = 2 53
PPCM (1000480250 )= 25 3 53 = 12000 :
-II :
(1 1
) : ( .
d a
b :
:
p gcd ( 249 )= 3
a 0 b 0 a 0 a + b 0b = d :
a 0 b 0 24a 0 + 9 b 0 = 3 :
a 0 = -1 b 0 = 3
(2 2
24 ( -1) + 9 ( 3)=3
) ( :
a b
b , a a a + b b = 1
:
* a b
1
p gcd ( a b )= 1 :
a
b
a a + b b = 1 :*
ab
a a + b b = 1
:
PGCD ( a b )= d d : a b
d :
d : a a b b
aa + bb
d : 1 d = 1 : a b
.
(3 :
-
b c
a
b.c
.
:
a
b
a
b
(1)...a a + b b = 1 : a c a b (2)...a a + b c = 1 :
(a a + b b )(a a + b c )= 1
(1) (2):
aa a 2 + ab ac + a b ab + bb bc = 1
:
(aa a + ab c + a b b ) a + ( bb )bc = 1
:
aa a + ab c + a b b = g a bc
( a a
.
b1 b2 ... bn :
n 2
bb = dg a + dbc = 1:
. :
bn
a
b1 ,b2,...,b n
b1 = b2 = ...= bn = b
.
(1 -III :
a b c
.
a bc a b a c
.
:
a bc
a
b
a a + b b = 1 :
c
q bc = a .q :
) (.
:
a ac + b bc = c
a
b
a ac + b .aq =c
:
:
ac + b q = p
:
a (a c + b q )= c
a .p = c a c
.
:
(2 :
( b
a1 a2 a1 a2
b
. a 1 a 2 :
a1 b
q = a1.q :
b
a2
a 1.q a2 a1 b a2
a2 q
b = a1.a 2.p :
p
:
q = a 2.p
a1.a 2
. b
( b
a1 , a 2,...,an
1
a1 a 2 ... a n
2
.n
:
396
2
b
22
2 11
.
:
x y 5 x = 7y :
.
:
5 x = 7y 5 y
:
3
x = 7k
:
:
k
7 5
x = 7k
y = 5k
7 5
:
y = 5k
y
5 x = 7 5k
k
:
2 (1)...5x - 3 y = 2 : (11)
. :
5 x - 3 y = 2 :::
5 (1) - 3(1)= 2
5 x - 3 y = 5 (1) - 3 (1)(1)...5 ( x - 1) = 3 ( y - 1)
3 ( y - 1) 5 3 5
5 : y - 1
y = 5k + 1 ::
k y - 1 = 5k :
(2) :
x - 1 = 3kx = 3k + 1 :
5 ( x - 1)= 3 5k
(1)
x = 3k + 1 : y = 5k + 1( x y)
. k
(3 : . :
PGCD ( 20015040)
:
PGCD ( 200150 ) = PGCD (10 2010 15)= 10 PGCD ( 2015)= 10 5 PGCD ( 43)= 50 1 = 50
PGCD ( 4050 ) =10 PGCD ( 45)= 10 1 =10
PGCD ( 20015040 )=10
:
(4 : :
a1 ,a 2,...,a n
. 1
(5 PGCD ( a b)PPCM ( a b) a b
a = da
:
x
:
a
:
a a = b b
b
a
x = aa
:
x = a da x = b db
:
a
b
.
PGCD ( a ,b )= 1
a b
b = d b
d
:
:
x = bb
a da = b db
b b a
q
x = b b = qa b = qa b d
::
b
b = q a
x = ( da b)q
m = da b
:
a b
m
:
:
m = da b = ab = b a a b
a b
m
m
.
m
a b
m = da b : m d = da dba b=m : m d = a b :dm = a b : d = 1 : (6 1
:
.
(7 2
:
.
(8 3
:
a ,b,l
:
PPCM ( l a lb ) = l PPCM ( a b ) :
PPCM ( l a l b ) PGCD ( l a lb ) = ( l a ) .( lb ) :::
PPCM ( l a l b ) l .PGCD ( a b ) = l ( l ab )PPCM ( l a l b ) PGCD ( a b )= l ab
:
:
l a bPGCD ( a b ))l PPCM ( a b ) PGCD ( a b= )PPCM ( la lbPGCD( a b )= )PPCM ( l a lb
:
PPCM ( l a lb ) = l .PPCM ( a b )
:
PPCM (100150 ) = 50 PPCM ( 23 )= 50 2 3 =300 (9 : .:
PPCM(150200350) :* PPCM (150200) :PPCM (150200 ) = PPCM ( 50 350 4)= 50 PPCM ( 34)= 50 3 4 = 600:
PPCM (150200350 ) = PPCM ( 600350)
:
PPCM ( 600350 ) = PPCM ( 50 1250 7)= 50 PPCM (127)= 50 12 7 = 4200
:
PPCM (150200350 )= 4200
:
:1 :
(1
8:lcm(
NUM
PPCM
(2
. .
(3 :
:2 :
(1
9:gcd(
NUM
PGCD
(2
. .
(3 :
:3
( x y)Ax + By = 1 :
.
:
1
.
A = 44 50B = 80 77C = 45 100 .
2
.
a
b a = 18459 :
b = 38093
.
a b PPCM ( a b)= 2226a + b= 148
4
.
5
8
.
.
(1 . 211 (2
6
x 2 - y2 = 211 :
.
x y :
7
.
:
d = PGCD ( x y)
m - 18d = 791
m = PPCM ( x y )
.
(1 PGCD ( 765 459 1683) (2
.
765 x + 459 y= 1683 :
.
(3 ( x y) x + y < 10 :8
.
.
2 52 x - 44 y= 92 :
(1
(2
PGCD ( x y)= d
.
.
.d
(3 ( x y) (4 ( x y) . - 10 < x < 40
9
.
(1 nr , U0 (2
( Un) -
U0r
r
S n = U 0 + U 1 + ... +Un
- .
q !2Pq = ( 2010) :
n10
U0 r
.
U0 r U02=U10 -U1
Pn = U 0 U 1 ... Un -
n
3 .5
.
- = 1,U0 = 3
(1
= 23
.d
:
Sn Pn n
3q 2[ 5] :
2Sn + 2 = 3q [ 5]
.
( x y ) 9 x - 14 y = 13 :
( 3 1)
.
(2 45x - 28 y = 130 :
( x y) - .
.
.
x 0 [ 2] y 0 [ 5]
N (3
5bb 6
2aa 3
9
7
.
a
b
N
.
11
.
(1 . 120
a (2
b
a= a + 4b b= 2a + 7b
.
PGCD ( a b ) = PGCD ( a b)
(3
x y
({
x + 4 y )( 2 x + 7 y) = 5880xy = 7m
12
.
:
m PPCM ( x y)
.
.
(1)...43x - 13y= l : l , y,x
.
(1 ( -3l - 10l ) . ( x y)
. (1)
N (2
:
b 0ggg
: -
a
a
13
b
b
ababa
g
:
g
N
5
.
43a - 13b = g
.
.
.
(1 ( x y)
:
(1)...5 x - 3 y=7
(2 ( x y)
6
.(1)
( x y ) : (3 ( x y) (1) PGCD ( x y). PGCD
.
14
.
a (1 b
-
.
a + b a b
.
(2 .84
(3
x y x + y= 84 :
m = d 2 m = PPCM ( x y) : d = PGCD ( x y)15 (1
.
993 170
.
(2 (1)...993 x - 170 y= 143 : x y . -
( x0 y0) (1) x 0 + y0 = 6 :
- ( x y) (3 a
.
.(1 )
a- 1
1986 340 14 300
.
:
1
.
A B C
:
A = 44 50 = 4 11 5 10 = 2 2 11 5 2 5A = 2 3 52 11B = 80 77 = 8 10 7 11 = 2 3 2 5 7 11B = 24 5 7 112C = 45 100 = 9 5 102 = 3 2 5 ( 2 5)= 32 5 22 52
C = 22 32 53PGCD ( A B C)= 22 5 = 20PPCM ( A B C)= 24 5 3 32 7 11= 1386000
2
.
PGCD ( 18459 3809) :3809 = 13 293 :
18459 = 3 2 7 293PGCD ( 18459 3809 )= 293
) (. 293
a
b 293
293 1
.
3
:
PGCD ( a b)= d
a = d .a
:
b = d b
d a b = 2226
:
:
.
d 2226= d a d b
:
d a + d b = 148 :
a b
d 2226= a b : :
.
a b :
a
b 293 1
.
a + b= 148
d ( a + b )= 148
d .148
148
:
148 = 22 37 : 148= 1 (1
: d
.148 74 37 4 2 1 :
a b = 2226 a + b = 148
a b
x 2 - 148 x+ 2226 = 0 : D = 3250 D ; 57 ,008 :
.
.
= 2 (2
: d
a b = 1113 a + b = 74
x 2 - 74 x+ 1113 = 0 :D = 256
a b
.
.
x1 = 37 - 16 = 21
x2 = 37 + 16 = 53
a = 21 b = 53 a= 2 21 = 42 :
b = 2 53 = 106
a = 53 b = 21 a = 106 : b=42 a b = 556 ,5 .: d = 4(3a+b=37 a b ; 60 ,1.: d = 37 (4 a + b = 4 a b ; 30 ,08.: d = 74 (5a+b=2 a b ; 15 ,04 .: d = 168 (6 a + b = 14
.
a1 a23
8
a7
. 1
3 23
:
27
a2 a1 :
:
128 :
8 3 23 24
.
24
.
5
.
(1 211 a2 209
:
a2
a
a2