iii 1 1 1 1................................ 1 2.......................... 3 3.............................. 5 4................................ 7 5................................ 9 6............................ 11 7............................. 13 8............................. 15 9............................. 17 10............................... 19 11........................... 22 12....................... 25 13..................... 27 14...................... 29 15...................... 31 16...................................... 34 17........................ 35

1 1 1 1 1. ( ) p p 2, 3, 5, 7,, p 1 Q = 2 3 5 7 p + 1 Q > p (i) Q p (ii) Q Q q Q q 2 p q > p p Q.E.D.

2 1 1 2. 4n 1 ( ) p 2, 3, 5, 7,, p Q = 2 2 3 5 7 p 1 4n 1 Q > p (i) Q p 4n 1 (ii) Q 4n 1 4n + 1 (4n 1 + 1)(4n 2 + 1) = 16n 1 n 2 + 4(n 1 + n 2 ) + 1 = 4N + 1 Q 4n 1 4n 1 q Q q 2 p q > p p 4n 1 4n 1 Q.E.D. 1. 3. 6n 1

2. 3 2 a, b gcd(a, b) lcm(a, b) gcd(a, b) = 1 a, b 4. a, b g = gcd(a, b) l = lcm(a, b) (1) a, b g (2) a, b l (3) ab = gl ( ) (1) (2) (3) a = ga 1, b = gb 1 gcd(a 1, b 1 ) = 1 l = ga 1 b 1 ab = g 2 a 1 b 1 = g(ga 1 b 1 ) = gl Q.E.D. 5. N, a, b, c (1) a, b N gcd(a, b) = 1 ab N (2) b ac gcd(a, b) = 1 b c (3) p ac p a, c ( ) (1) N a, b 4 (2) N l = lcm(a, b) gcd(a, b) = 1 l = ab ab N (2) b ac ac = bb gcd(a, b) = 1 b c b c (3) p a gcd(a, p) = 1 (2) p c Q.E.D.

4 1 1 6. 3 6 ( ) 3 f(n) = (n 1)n(n + 1) 5 (1) f(n) 6 = 2 3 f(n) 2 3 3 f(n) 2 3 3 f(n) 3 f(n) 6 2. 7. p 5 p 4 1 240

3. 5 3 x A x B x A x B A B A B B A A B x A x B x x A x B

6 1 1 8. x x 2 = 2 x ( ) 1, 2, 3, 4, 1, 4, 9, 16, 2 x x x = m n (n 1) x 2 = m2 n 2 (n 2 1) m2 2 n2 x Q.E.D. 3. 9. x x 2 = 3 x 4. x 10 x = 2 x

4. 7 4 x, y x y a x y mod a x a y a mod modulo 10. (1) (5) (1) x x mod a (2) x y mod a y x mod a (3) x y mod a y z mod a x z mod a (4) x x mod a y y mod a x + y x + y mod a (5) x x mod a y y mod a xy x y mod a ( ) (1), (2), (3) (4) x x = ma y y = na (x x) + (y y) = (m + n)a x + y x + y mod a (5) x y x y = (x+ma)(y+na) xy = (my+nx)a+mna 2 = (my+nx+mna)a xy x y mod a Q.E.D. 5. 11. x, y, z x 2 + y 2 = z 2 x, y, z 5

8 1 1 6. 12. x, y, z x 2 + y 2 = z 2 xyz 4 13. p a p a p 1 1 mod p ( ) a p a, 2a, 3a,, (p 1)a p ja ka mod p (1 k < j < p) (j k)a 0 mod p j k a p a, 2a, 3a,, (p 1)a p a 2a (p 1)a (p 1)! (p 1)! a p 1 1 mod p Q.E.D. 7. 14. p a a p a mod p 8. 15. p 2 5 1/p

5. 9 5 n n n (n 1) 2 1 n! n 0! = 1 n P r np r := n! (n r)! n r P Permutation 16. np r = n (n 1) (n r + 1) ( ) Permutation np r = n! n (n 1) (n r + 1) (n r)! = (n r)! (n r)! = n (n 1) (n r + 1) Q.E.D. np r 1 n n r n P r 9. 17. np r = n 1 P r + r n 1 P r 1

10 1 1 10. (1), (2) (1) 6 0, 1, 2, 3, 4, 5 4 0 4 300 156 (2) n 2 n + 1 0, 1, 2,, n r 0 r n np r 1 r r/2 n P r 1 + r (n 1) n 1 P r 11. 4 4 P 2 + 2 4P 3 + 4 P 4 = 84 2 A C B D

6. 11 6 n r n r n C r nc r := n P r r! n r C Combination 18. nc r = n C n r ( ) Combination nc r = n P r r! = n! r!(n r)! = n! {n (n r)}!(n r)! = n P n r (n r)! = nc n r Q.E.D. nc r 1 n n r n C r 12. 19. nc r = n 1 C r + n 1 C r 1 13. 20. (a + b) n a n r b r n C r

12 1 1 14. 0 (x + 1) 0 1 1 (x + 1) 1 x 1 1 2 (x + 1) 2 x 2, x 1 2 1 ( ) n (x + 1) n x n, x n 1, x n n r r 0 r 0 (x + 1) 0 1 1 (x + 1) 1 1 1 2 (x + 1) 2 1 2 1 3 (x + 1) 3 1 3 3 1 4 (x + 1) 4 1 4 6 4 1 n r 0 ( ) ( n n + 2 r r + 1 ) ( ) n + = r + 2 ( ) n + 2 r + 2

7. 13 7 20 (a + b) n a n r b r n C r., (a + b) n = n C 0 a n + n C 1 a n 1 b + n C 2 a n 2 b n + + n C n 1 ab n 1 + n C n b n.. 21. nc 0 + n C 1 + n C 2 + + n C n 1 + n C n = 2 n ( ), a = b = 1, ( ) = (1 + 1) n = 2 n ( ) = n C 0 + n C 1 + + n C n 1 + n C n,. Q.E.D. 22.. nc 0 + n C 1 2 + n C 2 3 + + n C n n + 1 = 2n+1 1 n + 1 ( ) r nc r = r n! r! (n r)! = n (n 1)! (r 1)! (n r)! = n n 1C r 1 n 1C r 1 r = n C r n, n C r 1 r = n+1 C r n + 1

14 1 1., ( ) = n C 0 + n C 1 2 + n C 2 3 + + n C n n + 1 = n+1 C 1 n + 1 + n+1 C 2 n + 1 + n+1 C 3 n + 1 + + n+1 C n+1 n + 1 = 1 n + 1 ( n+1c 1 + n+1 C 2 + n+1 C 3 + + n+1 C n+1 ) = 1 n + 1 {( n+1c 0 + n+1 C 1 + n+1 C 2 + + n+1 C n+1 ) n+1 C 0 } = 1 ( 2 n+1 1 ) n + 1 = ( ) Q.E.D. 15. nc 0 n C 1 + n C 2 + ( 1) n 1 nc n 1 + ( 1) n nc n = 0 16. 23. nc 1 + 2 nc 2 + 3 nc 3 + + n nc n = n 2 n 1

8. 15 8 a, d a, a + d, a + 2d, a + 3d, a, d. a, d a n a n = a + (n 1)d. 24.. a, d n S n S n = n{2a + (n 1)d} 2 ( ), S n = a 1 + a 2 + + +a n 1 + a n, S n = a n + a n 1 + + a 2 + a 1 S n = a 1 + a 2 + + a n 1 + a n +) S n = a n + a n 1 + + a 2 + a 1 2S n = (a 1 + a n ) + (a 2 + a n 1 ) + + (a n 1 + a 2 ) + (a n + a 1 ). a r+1 + a n r = a + rd + {a + (n r 1)d} = 2a + (n 1)d, 2S n = n{2a + (n 1)d}. 2,. Q.E.D.

16 1 1 17. ( ) ( ) ( ) ( ) 0 1 n n + 1 (1) + + + = = n + 1 0 0 0 1 (2) (3) ( ) 1 + + 1 ( ) 2 + + 2 ( ) n = 1 ( ) n = 2 ( n + 1 2 ( n + 1 3 ), (n 1) ), (n 2) (k 1)k (k 1)k(k + 1) (k 2)(k 1)k (3) = 2 6 18. ( ) n 2 = 1 ( ) n + 2 ( ) n + 1 2 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1

9. 17 9 a, r a, ar, ar 2, ar 3, a, r. 25.. a, r n S n a(1 r n ) S n = 1 r na (r 1 ) (r = 1 ) ( ) S n = a + ar + + ar n 2 + ar n 1, r = 1. S n = a } + a + {{ + a } = na n r 1, S n rs n = ar + ar 2 + + ar n 1 + ar n S n = a + ar + ar 2 + + ar n 1 ) rs n = ar + ar 2 + + ar n 1 + ar n (1 r)s n = a ar n, (1 r)s n = a(1 r n ). 1 r,. Q.E.D. 19. 26. x n 1 x 1.

18 1 1 20. n S n S n = 10n+1 9n 10 9 9, 99, 999, 9999, 21. n S n S n = 2 (10n+1 9n 10) 27 6, 66, 666, 6666, 22. x 1 1 + 2x + 3x 2 + + nx n 1 = 1 (n + 1)xn + nx n+1 1 x

10. 19 10 a 1 + a 2 + + a n ( ) a k k 1 n a k k i j 10 k = 1 + 2 + 3 + + 10 9 2 k 1 = 2 0 + 2 1 + 2 2 + + 2 8 8 (2j 1) = 1 + 3 + 5 + + 15 j=1 i 3 = 1 3 + 2 3 + 3 3 + + n 3 i=1 27. (1) (2) (3) (a n ± b n ) = a n ± b n ( ) ca n = c a n (c ) c = nc (c ) ( ) (1) (a n ± b n ) = (a 1 ± b 1 ) + (a 2 ± b 2 ) + + (a n ± b n ) = (a 1 + a 2 + + a n ) ± (b 1 + b 2 + + b n ) = a n ± Q.E.D. b n

20 1 1 23. 27 (2) (3) 28. (1) (2) ( ) (1) 24 k = 1 n(n + 1) 2 k 2 = 1 n(n + 1)(2n + 1) 6 k 1, 1 n k = n{2 + (n 1)} 2 = 1 n(n + 1) Q.E.D. 2 (2) (k + 1) 3 (k 1) 3 = 6k 2 + 2 { (k + 1) 3 (k 1) 3} = ( 6k 2 + 2 ) ( ) = (k + 1) 3 (k 1) 3 = { 2 3 + 3 3 + + (n 1) 3 + n 3 + (n + 1) 3} { 0 3 + 1 3 + 2 3 + + (n 1) 3} = n 3 + (n + 1) 3 1 = n 3 + (n 3 + 3n 2 + 3n + 1) 1 = 2n 3 + 3n 2 + 3n ( ) = 6k 2 + 2 = 6 k 2 + 2n

10. 21 2n 3 + 3n 2 + 3n = 6 k 2 + 2n k 2 = 1 6 (2n3 + 3n 2 + n) = 1 n(n + 1)(2n + 1) 6 Q.E.D. 24.. 1 + 3 + 5 + + (2n 1) = n 2 25.. 1 2 + 2 3 + 3 4 + + n(n + 1) = 1 n(n + 1)(n + 2) 3 26. 29. (k + 1) 4 (k 1) 4 = 8k 3 + 8k, k 3 = { } 1 2 n(n + 1) 2

22 1 1 11 n P(n) P(n) n (i) P(1) (ii) P(k) P(k + 1) (i) (ii) n P(n) n P(n) (i) (ii) P(1) (ii) P(2) P(2) (ii) P(3) P(3) (ii) P(4) P(4) (ii) P(5) (i) (ii) n P(n) (i) (ii) (i) (ii) n P(n)

11. 23 30. n n 3 + 2n 3. ( ) P(n) n 3 + 2n 3 n (i) n = 1 n 3 + 2n = 1 3 + 2 = 3 3 P (1) (ii) P(k) k 3 + 2k 3 k 3 + 2k = 3m (m ) P(k + 1) (k + 1) 3 + 2(k + 1) (k + 1) 3 + 2(k + 1) = (k 3 + 3k 2 + 3k + 1) + (2k + 2) = (k 3 + 2k) + (3k 2 + 3k + 3) = 3m + 3(k 2 + k + 1) = 3(m + k 2 + k + 1) (k +1) 3 +2(k +1) 3 P(k +1) (i) (ii) n P(n) Q.E.D. 27. 28. n 2n 3 3n 2 + n 6 n 5 n 2 n 3

24 1 1 31. n 2 n 2n ( ) P(n) 2 n 2n n (i) n = 1 P (1) = 2 1 = 2, = 2 1 = 2 P(1) (ii) P(k) 2 k 2k P(k + 1) 2 k+1 2(k + 1) P(k + 1) = 2 k+1 = 2 2 k 2 k 2k 2 2k = 2k + 2k 2k + 2 = 2(k + 1) = P(k + 1) (i) (ii) n P(n) Q.E.D. 29. n 5 n 2 n > n 2 30. 32. n 1 + 1 2 + 1 3 + + 1 n 2n n + 1

12. 25 12 33. ABC BC CA AB P Q R AP BQ CR S BP PC CQ QA AR RB = 1 A R S Q B P C ( ) ABS CAS S 1 S 2 ABS CAS BP PC = S 1 S 2 BCS S 3 ABS BCS CAS BCS CQ QA = S 3 S 1 AR RB = S 2 S 3 Q.E.D. BP PC CQ QA AR RB = S 1 S3 S2 = 1 S 2 S 1 S 3

26 1 1 31. 34. ABC BC CA AB P Q R BP PC CQ QA AR RB = 1 AP BQ CR 32. 35. ABC BC CA AB L M N AL BM CN 2 : 1 33. 36. ABC

13. 27 13 37. ABC A, B, C l BC CA AB P Q R A BP PC CQ QA AR RB = 1 l R Q B C P ( ) C AB l S PBR PCS BP : CP = RB : SC BP CP = BR SC QAR QCS AR : QA = CS : QC AR QA = CS QC Q.E.D. BP PC CQ QA AR RB = BP PC AR QA CQ RB = RB SC CS QC CQ RB = 1

28 1 1 A l R B Q C S P 34. 38. ABC BC CA AB P Q R BP PC CQ QA AR RB = 1 P Q R 35. 39. ABC A < B A BC P B CA Q C AB R P Q R

14. 29 14 40. ABCD AB CD + AD BC = AC BD D A C ( ) BD E B BAE = CAD D A E C ABE ACD B ABE = ACD ABE ACD AB : BE = AC : CD AB CD = AC BE (1.1) AED ABC AD : ED = AC : BC AD BC = AC ED (1.2) (1.1) (1.2) AD CD + AD BC = AC BE + AC ED = AC (BE + ED) = AC BD

30 1 1 Q.E.D. 36. 41. 1 1 + 5 2 37. 42. 2 a, b, c 1 a = 1 b + 1 c

15. 31 15 [ ] p 2 5 1 p 1 7 = 0.142857142857142857 142857 142 857 142 + 857 = 999 1 17 = 0.05882352941176470588235294117647 0588235294117647 05882352 35294117647 105882352 + 35294117647 = 99999999999 1 p [ ] p mod p mod 2

32 1 1 mod 3 mod 5 [ ] a 1, a 2, a 3,, a n,

15. 33 [ ] [ ] 1, 1, 2, 3, 5, 8, 13, 21, 34, n F n F n = F n 1 + F n 2 F 1 + F 2 = 1 + 1 = 2, F 4 1 = 3 1 = 2 F 1 + F 2 + F 3 = 1 + 1 + 2 = 4, F 5 1 = 5 1 = 4 F 1 + F 2 + F 3 + F 4 = 1 + 1 + 2 + 3 = 7, F 6 1 = 8 1 = 7 F 1 + F 2 + F 3 + F 4 + F 5 = 1 + 1 + 2 + 3 + 5 = 12, F 7 1 = 13 1 = 12

34 1 1 16 1. 4 2. a 2 + 1 3. n! + 1 4. p 2 p 1 2 p 1 5. n 2 2n + 1 2 2n + 1 6. 2 7. 1 n 1 + 2 n 1 + + (n 1) n 1 n n p 1 p 1 + 2 p 1 + + (p 1) p 1 p 8. n n 2 3n + 1 n 1 9. n 10. n 6 n

17. 35 17 1. p 2, 3, 5, 7,, p Q = 2 3 5 7 p 1 6n 1 p (6n 1 + 1)(6n 2 + 1) = 6N + 1 Q Q p 6n 1 2. 240 = 2 4 3 5 f(p) = p 4 1 = (p 1)(p + 1)(p 2 + 1) 2 4, 3, 5 (1) f(p) 3 p 3k + 1 f(3k + 1) = 3k(3k + 2)(9k 2 + 1) f(p) 3 p 3k 1 f(p) 3 p 5 p 3k (2) f(p) 5 p 5k + 1, 5k + 2, 5k + 3, 5k + 4 3 (3) f(p) 2 4 p 2k + 1 f(p) = 2 3 k(k + 1)(2k 2 + 2k + 1) k(k + 1) f(p) 2 4 3. 8 4. m, n x = m n 10m/n = 2 10 m = 2 n 2 m 5 m = 2 n 5. x, y, z 5 x ±1 mod 5 y ±1 mod 5 z ±1 mod 5 x 2 + y 2 z 2 0 mod 5 6. x, y, z 4 x 2 ±1 mod 4 y 2 ±1 mod 4 z 2 ±1 mod 4 x 2 + y 2 z 2 0 mod 4 xyz 4 7. a p a p a 1 a 2 p a = a 1 a 2 11 a p 1 a 1 mod p a p 2 a 2 mod p a p = a p 1 ap 2 a 1a 2 = a mod p a 11 8. 13 10 p 1 1 mod p n = (10 p 1 1)/p α = 1/p n = 10 p 1 α α α 10 p α

36 1 1 α 9. n 1P r = 10. (n 1)! (n r 1)! (n 1)! n 1P r 1 = (n r)! (1) 5 5 P 3 = 300 0 5 P 3 = 60 2, 4 2 (4 4 P 2 ) = 96 (2) n n P r 1 0 n P r 1 r (n 1) n 1 P r 11. 4 P 2 2 4 P 3 4 P 4 12. n 1C r = n 1 P r r! = (n 1)! r!(n r 1)! n 1C r 1 = n 1 P r 1 (r 1)! = (n 1)! (r 1)!(n r)! 13. (a + b) n = (a + b) (a + b) a n rb r n (a + b) r b ( ) n 14. n C r 19 ( n C r + n C r+1 ) + ( n C r+1 + r nc r+2 ) = n+1 C r+1 + n+1 C r+2 = n+2 C r+2 15. 18 16. S = 0 nc 0 + 1 nc 1 + 2 nc 2 + + n nc n S = n nc n + 1 nc 1 + 0 nc 0 2S = n( n C 0 + 1 nc 1 + + n nc n ) 15 2S = n 2 n ( ) k 17. (1) = 1 ( ) 0 k n(n + 1) (2) = k 1 + 2 + + n = ( 1) 2 ( ) ( ) k (k 1)k (k 1)k(k + 1) (k 2)(k 1)k 2 3 (3) = = S = + + ( 2) 2 6 2 2 n + = 1 2 3 + 2 3 4 1 2 3 (n 2)(n 1)n (n 1)n(n + 1) + + + 2 6 6 ( 6 ) (n 1)n(n + 1) (n 2)(n 1)n (n 1)n(n + 1) n + 1 S = = 6 6 3 ( ) ( ) n n + 1 18. + = n 2 2 2 19. 25 (x 1)S n = x n 1 20. S n = (10 1) + (10 2 1) + + (10 n 1) = 10 + 10 2 + + 10 n n

17. 37 S n = 10(10n 1) 9 21. S n = 2 3 10n+1 9n 10 9 n 6 + 66 + 666 + = 2 (9 + 99 + 999 + ) 18 3 22. S = 1+2x+3x 2 + +nx n 1 xs = x+2x 2 +3x 3 + +nx n (1 x)s = 1+x+x 2 + +x n 1 nx n 19 (1 x)s = 1 xn 1 x nxn 23. 27 24. ( ) = (2k 1) = 2 k = 2 1 2 n(n + 1) n = n2 = ( ) 25. ( ) = k(k + 1) = = 1 n(n + 1)(n + 2) = ( ) 3 26. k 2 + { (k + 1) 4 (k 1) 4} =, 28 (2) 6n 2 + 4n + 1, k = 1 6 n(n + 1)(2n + 1) + 1 n(n + 1) 2 (8k 3 + 8k) { (k + 1) 4 (k 1) 4} = 2n 4 + 4n 3 + (8k 3 + 8k) = 8 k 3 + 4n(n + 1) 27. (i) n = 1 2n 3 3n 2 + n = 2 3 + 1 = 0 6 (ii) n = k 2n 3 3n 2 + n = 2k 3 3k + k 6 2k 3 3k 2 + k = 6m (m ) n = k + 1 2n 3 3n 2 +n = 2(k+1) 3 3(k+1) 2 +(k+1) = 2(k 3 +3k 2 +3k+1) 3(k 2 +2k+1)+k+1 = 2k 3 + 3k 2 + k = 2k 3 3k 2 + k + 6k 2 = 6m + 6k 2 = 6(m + k 2 ) n = k + 1 2n 3 3n 2 + n 6 (i) (ii) n 2n 3 3n 2 + n 6 28. (i) n = 1 5 n 2 n = 5 2 = 3 3 (ii) n = k 5 n 2 n = 5 k 2 k 3 5 k 2 k = 3m (m ) 5 k = 2 k +3m n = k +1 5 n 2 n = 5 k+1 2 k+1 = 5 5 k 2 2 k = 5(2 k + 3m) 2 2 k = 5 2k + 15m 2 2 k = 3 2 k + 15m = 3(2 k + 5m)

38 1 1 n = k + 1 5 n 2 n 3 (i) (ii) n 5 n 2 n 3 29. (i) n = 5 2 n = 2 5 = 32 n 2 = 5 2 = 25 2 n > n 2 (ii) n = k 2 k > k 2 n = k + 1 2 k+1 (k+1) 2 = 2 2 k (k 2 +2k+1) > 2k 2 (k 2 +2k+1) = k 2 2k 1 = k(k 1) 1 > 0 2 k+1 > (k + 1) 2 n = k + 1 2 n > n 2 (i) (ii) n 5 n 2 n > n 2 30. (i) n = 1 ( ) = 1 2 1 = 1 ( ) = 1 1 + 1 = 1 (ii) n = k 1 + 1 2 + 1 3 + + 1 k 2k k + 1 n = k + 1 1 + 1 2 + 1 3 + + 1 k + 1 k + 1 2k k + 1 + 1 k + 1 = 2k + 1 k + 1 ( ) 2k + 1 k + 1 2k + 1 k + 1 2(k + 1) (k + 1) + 1 2(k + 1) (2k + 1)(k + 2) 2(k + 1)2 = (k + 1) + 1 (k + 1)(k + 2) = k (k + 1)(k + 2) > 0 2k + 1 k + 1 2(k + 1) > ( ) (k + 1) + 1 1 + 1 2 + 1 3 + + 1 k + 1 k + 1 2k + 1 2(k + 1) > k + 1 (k + 1) + 1 n = k + 1 (i) (ii) n 31. BQ CQ S AS BC P AP BQ CR S 33 BP P C CQ QA AR RB = 1 BP PC = BP P C P P BC P P AQ BQ CR 32. 34 AL BM CN G AG BC AG = GD D ABD NG BD GC BD ADC MG CD GB CD GBDC GD BC L

17. 39 AG : GL = 2 : 1 BG : GM = 2 : 1 CG : GN = 2 : 1 33. A, B, C BC, CA, AB I, J, K ABI CBK AB : BI = CB : BK BI BK = AB BC CJ CI = BC AC AK AJ = AC BI AB BK CJ CI AK AJ = AB BC BC AC AC AB = 1 34 34. P BC PQ AB A, B R 37 BP PC CQ QA AR R B = 1 AR RB = AR R B R R AB R R P Q R 35. RC S AS BC AC C D ACS = RCD = BCB = CSA AC = AS RAS RBC RA : RB = AS : BC = AC : BC R AB CA : CB AR : RB = AC : BC AP A BP : PC = AB : AC BQ B CQ : QA = BC : AB AR RB BP PC CQ QA = AC BC AB AC BC = 1 38 AB P Q R 36. ABCDE ABCD 40 AB CD + AD BC = AC BD x = DA = AC = BD 1 + x = x 2 x = 1 + 5 2 37. ABCDEFG a = AB b = AC c = AD ACDE 40 AC DE + CD EA = AD CE b a + a c = c b abc 1 a = 1 b + 1 c [ ] 1 2m p 99 9 8

40 1 1 102m p 1 p 102m 1 mod p 10 m 1 mod p 10m p + 1 p 10m p 1 1 = 0.99 99 9 p [ ] mod 3 n n = 0, 1, 2, 3 k n n a 1, a 2, a 3,, a s n 3 k a 1, 0,, 0, a 2, 0,, 0, a 3, 0, 0,, 0 0, a s a i a i+1 0 3 k 1 n 3 k n O 3 k 1 n (x + 1) n = x n + a 1 x n 1 + + a s 1 x + 1 (x + 1) 3 x 3 + 1 mod 3 a 3 i a i mod 3 (x + 1) 3n = x 3n + a 1 x 3(n 1) + + a s 1 x 3 + 1 3n n O 2 3 k n n O 3 k 1 [ ] 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 2 1, 0, 1, 0, 3 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 5 1, 4, 4, 1, 0, 1, 4, 4, 1, 0, 1, 4, 4, [ ] A,B,C,D AB = CD = 10 AD = 5 2 BC = 2 AC = BD = 2 5 AB CD + AD BC = AC BD = 20 A,B,C,D [ ] (1) F 1 + F 3 + F 5 + + F 2n 1 (2) F 2 + F 4 + F 6 + + F 2n

17. 41 (3) F 1 F 2 + F 3 + + ( 1) n+1 F n (4) F 1 F 2 + F 3 + + ( 1) n+1 F n (5) F 2 1 + F 2 2 + F 2 3 + + F 2 n