- : 1, 10 , . :

1() 3..() 4 :1. ..() 5-62. ..() 7-83. ..() 9-134.- ..() 145. .() 15-19 :1. .() 20-212. .....() 22-233. .....() 24-254. () 26-275. .() 286. ..() 29-307. ..() 318. ..() 329. ......() 3310. .() 34() 35 () 36

9 - , 1, , , . . -, , 9- , . -, . , . -, - , , . , : . , . , II .

, Microsoft Office Power Point 2007. : , - 36 , , , , , . : , , , - , . : , , , , , , , , , . , . , .

: , =b*, b ( b ). - b.: b ( b)

N, b N, N1) b, - - 2) , b b, ...3) b, b c a c4)a b, b a=b a=-b5)a b, b > a a=06)a b, a 0 a b 7) b, , b

1

: N, b N, N 8) 1 b, a2 b an b (a1a2 an) b9) (a1+a2++an) b a1 b, a2 ban-1 b an b10)a b a>0 a b11)a b, b c, m N0, n N0, ma>nb, mo(ma-nb) c (ma+nb) c12)a b, k 0 ak bk 13)ak bk, k 0 a b14)a bc (a b) c15)(a b) c a bc

: >1 , : 1 . : m>1 , , , , 1 m. :1)2 2)a b ab a b* b a* (, - )3) ,b,c,d Z bcd , - b c d 4)a N, >1 -

7 a N, 1, 1, 2, 3,,k = 1*2*3**k

1, 2, 3,,k = 1 1*2 2*3 3**k k

: 1, 2n , .: (a1, a2an)=d, d- 1, 2n a) d>0b) d a1, d a2 d an 1:1) 1, 2n, , 2)p1,,ps a1=11**ss,,an=p11**pss (1, 2n)=p1min(1,,1) **psmin(1,,s): :1) , 2)

9 1: 10080, 2646, 56.:1)10080 2 2646 2 56 2 5040 2 1323 3 28 2 2520 2 441 3 14 2 1260 2 147 3 7 7 630 2 49 7 1 315 3 7 7 105 3 1 35 5 7 7 1 2)d= 21*30*50*71= 2*7=14 (10080,2646,56)=14

10080=25*32*5*7 =25*32*51*71 2646=2*33*72 =21*33*50*72 56=23*7 =23*30*50*71 2: (a1, a2an)=d, d b, b>0 ( ,, )= 3: (1,,an-1,an)=((a1,,an-1),an)n3 n-: 1) (n-1)2) (d, an), d= (a1, a2an), an -

: a1, a2,,an , . : [a1, a2an]=m, m- a1, a2an)m>0b)a1 m,,an m

11 5: a1=11**ss*,,*an=p11**pss - m=[a1, a2an]=p1max(1,,1) **psmax(1,,s) 6: >0, b>0, a N, b N, (a,b)=d, [a,b]=mm=

: :1) , 2) - ,

12 1: 96,64,33,22.: 1)96 2 64 2 33 3 22 2 48 2 32 2 11 11 11 11 24 2 16 2 1 1 12 2 8 2 6 2 4 2 3 3 4 2 1 2 2

2)m=26*31*111=2112[96,64,33,22]=2112

96=25*3 =25*31*110 64=26 =26*30*110 33=11*3=20*31*111 22=11*2=21*30*111

- : b - , (a,b)=1 1: Z, Z, - - 2: ,b - [,b]=ab 3: :b : : b 4: (* b) , (,)=1 b

14

: ax+by=c1. d(, b)2. , , , (0; 0)- .3. : =-bk+x0, y=ak+y0, k Z

1: -3=15: a)(1;3)=1b) : =(15+3):1 =0, =(15+0). , (15;0) - c) : =3k+15, k Z y=k+0=k, k Z: (3k+15; k), k Z

2: 15+11=14:)(15;11)=1b) : =(14-11):15 , [0;14], .. (;), , (14-11):15 ( ). (-2;4) c) : =-11k-2, k Z y=15k+4, k Z: (-11k-2; 15k+4), k Z

3: 390 7 12 . ?: ) 7 , - 12. (7+12) , 390. . 7+12=390b)(7;12)=1c) : =(390-12):7 , [1;6] (54;1) d) : =-12k+54, y=7k+1 k Z

k0123454423028618152229,

: , , , . 1: : m2- n2=2001. : (m-n)(m+n)=20012001=3*23*29*1

: (1001;1000), (335; 332), (49; 20), (55;32)

m=1001n=1000m=335n=332m=49n=20m=55n=32 m-n=1 m+n=2001m-n=3 m+n=667m-n=23m+n=87m-n=29m+n=69 2: 2-3+22=3: : 2-2-+22=3; (2-)-(2-22)=3 : (-)-2(-)=3; (-)(-2)=3 4 : 1) 2) 3) 4)

( 2 ) :(5:2); (1:2); (-5:-2); (-2:-1);-=3-2=1-=-1-2=-3-=-3-2=-1-=1-2=3=+3 +3-2=1=5=2=1=2=-1 -1-2=-3, 1: 19+89=1989: 19+89=1989 19-1900=89-89 19(-100)=89(1-) (*)(19;89) - (*) 3 ) -100=89 b) -100=-89 c) -100=0 1-=19 1-=-19 1-=0

) = b) =11 c) =100 =20 =1: (11;20), (100;1)

2: 2-22=1 : 22- - 22=2-1= (-1)(+1) (-1) : 2(.. ) (+1): 2(.. ) - ,-

: (3;2)

=2

=3(-1)(+1):4

23

1: 2-+5-9=0:

) , b) Z, = 1, 3, 9

=-1, =13 =3, =5=1, =-3, =-9, =-3=-3, =5 =9, =13 (-1;13);(1;-3);(-3;5);(3;5);(-9;-3);(9;13).

, 9x2+5-9==+5-9x Zx

, 2: --=2: ) : (-)=2+ (1-)=2+ = =-1-b).. Z; Z, (-1) 1; 3, =2, =-4,=0, =2,=4, =-2,=-2, =0.: (-2;0);(0;2);(2;-4);(4;-2)

+2 1- 3 -1

, 1: , 2++1 + 2++1 = 13:)2++1=(+1)+1 (+1)- 2++1 - b) 2++1 - ) : .+.=. .+.=.( )

(+1)+1 -

, 2: 3+3-3=2: 1) , 3- 3- 3- : +- 2) -, - 3- 3- 3- : +- (, -, -)3) -, -, =2m, y=2n 8m3+8n3-12mn=2 2(2m3+2n3-3mn)=1 m n: 1: : 2(4-22+3)(4-32+4)=7 (1): 4-22+3=4-22+1+2=(2-1)2+22 4-32+4=(2-3 )2+7 7 .. 7, ..=7,, (1) : (2-1)2+2=2 2-1 =0 (2-3 ) +7 = 7 2- =0

=1, = Z: . ( )2444422

233

1 : n!+4n-9=k2: , n!+4n-9=n!+4n-12+3) n 4, (n!) 4, 4n 4, 12 4 (4(n!+4n-9)=3) k, 4 3. n 4 .b) , n=1,2,3 : 1.n=1 2.n=2 3.n=3 1+4-9=k2 2!+8-9=k2 3!+12-9=k2 -4=k2 1=k2 9=k2 k= k=1 k=3: n=2, k=1 n=3,k=3

2 : 2+1=3: ) 3 3, b) (2+1)/3: (3(2+1)=0), (3(2+1)=1), (3(2+1)=2) 1.=3k (3(9k2+1)=1) 2.x=3k+1 (3(9k2+6k+1+1)=2) 3.x=3k+2 (3(9k2+12k+4+1)=2): (2+1) 3 3 3

:

, 1: 54+102+26+43 =6: 54+102+26+43 = 5(4 +22)+2(6+23) = 5(2+1)2+2(3+1)2-7 : 5(2+1)2+2(3+1)2=13

5(2+1)2 13 (2+1)2 , (2+1) 0,1,-1

, =0 5*1+2(3+1)2=13 (3+1)2=4 , 3+1=2, 3+1=-2, =-3 ( ) 3+1=2;=1

:(0;1)

1: 54+102+26+43 =6.: 54+102+26+43 = 5(4 +22)+2(6+23) = 5(2+1)2+2(3+1)2-7 : 5(2+1)2+2(3+1)2=13 5(2+1)2 13 (2+1)2 , (2+1) 0,1,-1 , =0 5*1+2(3+1)2=13 (3+1)2=4 , 3+1=2, 3+1=-2, =-3 ( ) 3+1=2;=1

:(0:1)

1 : 3-100=225: , 3 5 = 5z, z Z, 125z3-100=225y 5z3-4=9y (1), 9,.a) z=3t b) z=3t+1 c) z=3t-15(3t)3-4=9y 5(3t+1)3-4=9y 5(3t-1)3-4=9y135t3-4=9y 5(27t3+27t2+9t+1)-4=9 5(27t3-27t2+9t-1)-4=9y 135t3+135t2+45t+1=9y 135t3-135t2+45t-9=9y.. z=3t-1, =15t-5, y=15t3-15t2+5t-1: (15t-5; 15t3-15t2+5t-1), t Z

, 1: : 7(+)=3(2-+2): +=, -=q. , , : = p+q , =p-q . : 7= - - 7= ..28p=3(p2+3q), p p 3, . p=3k, k Z p=3k, 28*3k=3((3k)2 +3q2); 28k=3(3k2 +q2). , k 3, k=3m, m Z; k=3m, 28*3m=3(3(3m)2 + q2; 28m=27m2+q2 ; m(28-27m)=q2; q20, m=0, m=1 ( m(28-27m) 0 c ) ) m=0, k=0 (.. k=3m), p=0 (.. p=3k), q=0(.. 28p=3(p2+3q)), , =0, =0 (.. x=p+q , = p-q b) m=1, k=3, p=9, q2=1(.. m(28-27m)=q2) )q= 1, =5; =4; b) q= -1, =4; =5;:(5:4);(4:5);(0:0) -

22(+q)2 (p+q)(p-q) (p-q)2

2223(p2+2pq+q2-p2+q2+p2-2pq+q2)

4

22

1 : 10+=2+2+13: 10+=2+2-13 2-10+2-+13=0 D/4=25-y2+y-13 D/40, ..25-2+-13 0-2++12 0 *(-1)2--120D=1-4*(-12)=49=72y1=-3y2=4 + + -3 - 4 .. -3 4, .. : =4, 3, 2, 1,0,-1,-2,-3: (-5;-3), (5;4)

- . . . , , . , , .. , 9 11. , , , . , , , , , . , , , , .

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