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