7
1.
1. n, n2+1 n+1.
2. , .
3. , 4p2+1 6p2+1 .
4. a b, (a,b)=24; [a,b]=2496.
5. , 1476 41.
6. 109348 14.
7. n!, n=50.
8. 20x 10(mod 25).
9. 45x-37y=25.
10. : .
11. 0,0001 .
2.
1. , , , 8 1.
2. , n 3(15+25+...+n5) (13+23+...+n3).
3. , .
4. a b, a+b=667; [a,b]=120(a,b).
5. , 300 20.
6. 439291 60.
7. n!, n=123.
8. 15x 3(mod 45).
9. 81x-48y=33.
10. : .
11. 0,0001 .
3.
1. , 9.
2. , mn+pq m-p, mq+np m-p, m,n,p,q .
3. , , p+10 p+14 - .
4. a b, a∙b=720; (a,b)=4.
5. , 1665 37.
6. 243402.
7. N, 5832.
8. 78x 42(mod 51).
9. 17x-25y=117.
10. : .
11. 0,0001 .
4.
1. , .
2. , a2+ b2 7, a b , 7 b 7.
3. , , a+m, a+n , a>3 m n 3 , 1 2.
4. a b, a+b=144; (a,b)=24.
5. , 1476 41.
6. 100!?
7. x, ()=12.
8. 39x 84(mod 93).
9. 43x-37y=21.
10. : .
11. 0,0001 .
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5.
1. , , - , 2, 4.
2. , 9.
3. , 2+1, - , .
4. 667, 120. .
5. , 975 13.
6. 2100.
7. n, n τ(n)=6, a σ(n)=42.
8. 12x 51(mod 39).
9. 17x-16y=31.
10. : .
11. 0,0001 .
6.
1. , , , 1 .
2. , 3k+2, kÎZ .
3. , =6k-1.
4. a b, , (a,b)=45.
5. , 1072 8.
6. 439291 60.
7. N, 330×540.
8. 20x 10(mod 25).
9. 53x-17y=25.
10. : .
11. 0,0001 .
7.
1. , (n3-1):7, (n3+1):7, n 7, nÎZ.
2. , n .
3. , p q - , 3, p2-q2 24.
4. a b, [a,b]=975.
5. , 2476 619.
6. 9341 111.
7. n, n τ(n)=12, a σ(n)=465.
8. 5x 26(mod 12).
9. 23x-15y=19.
10. : .
11. 0,0001 .
8.
1. n, n2+1 n+1.
2. , n 3(15+25+...+n5) (13+23+...+n3).
3. , p+10 p+14 - .
4. a b, , (a,b)=28.
5. , 975 13.
6. 439291 60.
7. 29x 3(mod 12).
8. N, 5832.
9. 41x-114y=5.
10. : .
11. 0,0001 .
9.
1. , mn+pq m-p, mq+np m-p, m,n,p,q .
2. , , 4 1.
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3. , .
4. a b, (a,b)=24; [a,b]=2496.
5. , 1072 8.
6. n, n=pq, p q - , , p-q=2 φ(n)=120.
7. 16x 50(mod 23).
8. 47x-105y=4.
9. 2153.
10. : .
11. 0,0001 .
10.
1. , (22n-6):10, - n≥2.
2. p ≥5. , (2-1):24.
3. , .
4. a b, ab=168; (a,b)=14.
5. , 975 13.
6. , 295!
7. n, n 12 τ(n)=14.
8. 15x 3(mod 45).
9. 17x-16y=31.
10. : .
11. 0,0001 .
7
1. , n(n2+5), n- , 6.
.
, , : n(n2+5)=n*[(n-1)(n+1)+6]=
=(n-1)*n*(n+1)+6n/
6, k k!( , Cnk= - ), (n-1)n(n+1) 6 - n.
2. , .
. : (n-2);(n-1);n;(n+1);(n+2). , (n-2)2+(n-1) 2+n2+(n+1) 2+(n+2) 2=5(n2+2) , , (n2+2) 5, n2 8 3, . , .
3. , a, a+m, a+n , a>3 m n 3 , 1 2.
. >3; m=3t+1; n=3t1+2. , 2 3, p=6k 1. a=p=6k+1, a+n=6k+1+3t+2=3(2k+t1+1)- . a=p=6k-1, a+m=6k-1+3t+1=3(2k+1) . , : a, a+m, a+n , a>3 m,nN 3 1 2 .
4. a b, (a,b)=24;[1,b]=2496.
. (a,b)=24, a=24m b=24n, (m,n)=1. m<n. [a,b]= , 2496= , m*n=104=23*13. (m,n)=1, m*n=1*104 m*n=8*13. : m=1 n=104 , m=8 n=13
5. , 300 20.
. (300,)=20 300. ϳ 20, (15,)=1, 15 15. ʳ : f(15)=8. : =1,2,4,7,8,11,13,14, =20,40,80,140,160,220,260,280.
6. , 295!.
. , . ij n!=paqbrc, , , m, m- k s, k s 2 5 n!. n! 5 2, s, 5 n!. s s=[ ]+[ ]+[ ]+ [ ]=[ ], : s=[ ]+[ ]+[ ]=59+11+2=72. , 295! 72 .
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7. 2232123 52.
. al m, (l,m)=1 s f(m), s ( ): s=f(m)q+r, 0 r f(m). f(m)=1(mod m), s= f(m)q+r= f(m)q r= r(mod m), r , s. : 52=22*13;f(52)=
22*13(1- )(1- )=24. 223=52*4+15; 2123=24*88+11; 2232123= =(52*4+15)24*88+11 1511= 159*152=(153)3*225 33752*17 (-5)3*17=
= (-125)*17 =-357 7(mod 52). , 2232123 52 7.
8. n, 5832.
. p=5832, p=23*36. n=2a*3b : p= . , =23*36, a(a+1)(b+1)=6; b(a+1)(b+1)=12. a=1; b=2, n=2*36=18.
9. : 20 10(mod 25).
. (20,10,25)=5, 5, : 4 2(mod 5), 2x=1(mod 5). : x 3(mod 5), xk+1 m1k+a(mod m), : x1 3; x2 8; x3 13; x4 18; x5 23(mod 25). : 20*3-10=50 25; 20*8-10=150 25, 20*13-10=250 25; 20*18-10=350 25, 20*23-10=450 25.
10. : -117+343=119.
. : 117(-)+343=119. . ax+by=c, a,b,c- ; (a,b)=1, : =(-1)n-1cQn-1+bt; y=(-1)ncP n-1-at, t- , Q n-1 Pn-1- . =117; b=343; (117;343)=1. : =[0;2,1,13,1,1,1,2]. , n=7. Pn-1= P6; Qn-1= Q6; P6= 44; Q6=129. 0=(-1)6*119*129=15351, 0=(-1)7*119*44=-5236. =15351+343t, y=-5236-117t, =-15351-343t,
y=-5236-117t. 0,0 . , . t=45. x1=84; y1=29 x=84+343k; y=-29+117k ( t k).
11. .
. 3587:2743=1+ , a0=1; 2743:844=3+ , a1=3; 844:211=4, a2=4; =1+ ; , .
12. 0,0001 .
. , : ; ; ; ; ; ; a5=a1, . . :
k | -1 | |||||||||
qk | - | |||||||||
Pk | ||||||||||
Qk |
. , 0,0001 , .
13. .
. 243402 100. . (43,100)=1, , . , 4 9.
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8.
I. ` :
1. ,
2. (mod 5),
3. (mod 7),
4. (mod 11),
5. (mod 11),
6. (mod 3),
7. (mod 5),
8. (mod 5),
9. (mod 5),
10. (mod 5).
II. ` , :
1. (mod 5),
2. (mod 17),
3. (mod 31),
4. (mod 41),
5. (mod 47),
6. (mod 13),
7. (mod 23),
8. (mod 5),
9. (mod 7),
10. (mod 7).
III. :
1. 5;
2. 7;
3. 11;
4. 13:
5. 17;
6. 23;
7. 37;
8. 53;
9. 19;
10. 43.
IV. a m, :
a | ||||||||||
m |