2012 A, N, Z, Q, R, C

2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1

1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2

N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii) N { 5, 6, 7, } = {a a 5 } { } { } 1 2, 1 4, 1 1 8, = k 2 k (iii) N A 1 A x A x + 1 A A N n n = 1 n = k n = k + 1 (iii) p(n) n n p(n) 3

(1) p(1) (2) p(k) p(k + 1) (3) (1), (2), n p(n) (iii) p(n) n N A A = {n n N, p(n) } (1) 1 A (2) k A k + 1 A (iii) A = N n p(n) ( ) N a b a + b ab N a(x + y) = ax + ay N N Z Z Q 4

2 2.1 0 Z Z = {, 3, 2, 1, 0, 1, 2, 3, } a a + x = a x 0 0 a + x = 0 x a Z 0 0 Z Z ( ) 12 3 Z 12 5 Z 75 6 6 75 6 75 1 a b 0 a = qb + r, 0 < = r < b q, r q a b r b a, 3b < = n < 2b, 2b < = n < b, b < = n < 0, 0 < = n < b, b < = n < 2b, a b q qb q qb < = a < (q + 1)b q q r = a qb qb < = a < (q + 1)b 0 < = r < b a = qb + r q, r a = q 1 b + r 1, a = q 2 b + r 2, 0 < = r 1, r 2 < b 5

0 = (q 1 q 2 )b + r 1 r 2 r 1 r 2 b 0 < = r 1, r 2 < b b < r 1 r 2 < b b 0 r 1 r 2 = 0 q 1 = q 2 ( ) 2.2 2 a b a = bq q b a. a b a b 0 6 2 3 2 3 6 = ( 2)( 3) ±1 1 ±1 Z 1 1 3 a, b, c, 0 a, b, c, (least common multiple L.C.M.) a, b, c, 1 a, b, c, (greatest common measure G.C.M. greatest common divisor G.C.D.) 1 2 a b (a, b) (12, 32) = 4 (1) (2) 6

2 (1) (2) (3) a, b l d ab = dl (4) a, b c b bc a c a - (1) a, b, c, l, m m l q r m = ql + r, 0 < = r < l l m a l = al, m = am r = m ql = a(m ql ), r a b, c, r a, b, c, r 0 l r, l r = 0 m l ( ) 2.3 24 42 2) 24 42 3) 12 21 (24, 42) = 6 4 7 6188 4709 3 a > b > 0, a b r (a, b) = (r, b) a b q a = bq + r (a, b) = d 1 a = a d 1, b = b d 1. r = a d 1 b d 1 q = d 1 (a b q) r d 1.,d 1 b r d 1 < = (b, r) = d 2 (b, r) = d 2, b = b d 2, r = r d 2. a = b d 2 q + r d 2 = d 2 (b q + r ) d 2 < = (a, b) = d 1 d 1 = d 2 (a, b) = (b, r) ( ) 0 < = r < b 7

1 (6188, 4709) 6188 = 4709 1 + 1479 4709 = 1479 3 + 272 1479 = 272 5 + 119 272 = 119 2 + 34 119 = 34 3 + 17 34 = 17 2 (6188, 4709) = (4709, 1479) = (1479, 272) = (272, 119) = (119, 34) = (34, 17) = 17 2 2 ( Euclid) Eukleides 365? 275? 1 1 3 13 1 6 7 9 10 11 13 2.4 1 1 10 3 a, b (1) ab 3 a b 3 (2) a + b ab 3 a b 3 (3) a + b a 2 + b 2 3 a b 3 10 2 +(0)+(4) p 3 a, b (0) m, n mn p m n p. 8

(1) a + b ab p a b p (2) a + b a 2 + b 2 p a b p (3) a 2 + b 2 a 3 + b 3 p a b p (4) n a n + b n a n+1 + b n+1 p a b p 2 2 98 n 8 r(n) (a, b) 0 < a r(a) < 4 3 r(b), 0 < b r(b) < 4 3 r(ab) (1) a r(a) r(b) (2) a b 3 3 06 1 (1) k k k + 1. (2) n 3 n 2 k n 2 = 2k + 1 n, k, k + 1. (3). 4 4 98 a, b, p, q p 2 + q 2 a = pq b a b 1 (1) pq b (2) a + 2b 5 5 [09 ] n (1) (n 2 + 1) (n + 2)(n 2) = 5 n + 2 n 2 + 1 1 5 (2) (1) n + 2 n 2 + 1 1 n 9

(3) (1),(2) 2n + 1 n 2 + 1 1 n. 6 6 98 k, l (k > = l) {a n }, {b n } a 1 = k, b 1 = l n > = 1 { { b n (b n 0 ) a n b n (b n 0 ) a n+1 = b n+1 = (b n = 0 ) (b n = 0 ) a n (1) k = 1998, l = 185 {a n }, {b n } 5 (2) k, l, n b n > = b n+1 ( b n = 0 ) (3) k, l b n = 0 n (4) b n = 0 n a n k l b n 10

3 3.1 0 ±1 a ±1 ±a ±1 ±a 4 a α e α = ±p 1 e2 e 1 p 2 p m m p 1, p 2,, p m e 1, e 2,, e m 48 ( ) 48 = 2 2 12 = 2 2 2 2 3 = 2 4 3 48 = 3 16 = 3 2 4 4 () (1) 4 4 = 2 2 (2) a ( ) a a a = b c (1 < b < a, 1 < c < a) b c a a a a = p 1 p 2 p m = q 1 q 2 q n 2 (4) p abc = (ab)c p p 1, p 2,, p m q 1 p 1 q 1, p 1, p 1 = q 1 p 2 p m = q 2 q n b b < a a (3) (1)(2) ( ) 11

2 (1) (2) (1) 1 (2) (1) a b p ab p a b p (2) 2 (1) a p a p ab p 2 (4) b p b p a p ( ) (2) 2 = n m 2m 2 = n 2 n 2 2 2m 2 2 2 ( ) 5 n p 1, p 2,, p n a = p 1 p 2 p n + 1 4 a a p 1, p 2,, p n 1 p 1, p 2,, p n a a p 1, p 2,, p n p 1, p 2,, p n ( ) 3.2 x 2 + y 2 = z 2 (x, y, z) x, y, z 1 x, y z 1 x y y. x z. y 2 = (z x)(z + x) 12

x z z x, z + x. ( ) 2 y = z + x z x 2 2 2 1. y 2, z + x, 2 z + x, 2 z x 2 z x 2 2 d, z + x = 2du, z x = 2dv, z = d(u + v), x = d(u v), y 2 = 4duv x, y, z 1. z + x, 2 z x 2 1 z + x z x, 2 2 z + x z x, z + x z x, 2 2 2 2 z + x z x, 2 2 z + x = 2a 2, z x = 2b 2 x = a 2 b 2, y = 2ab, z = a 2 + b 2 1 a b a b a > b x, z a b (x, y, z) 1. 1 (x, y, z) = (a 2 b 2, 2ab, a 2 + b 2 ), (2ab, a 2 b 2, a 2 + b 2 ) a > b, (a, b) = 1, a b... 2. 2 a b (x, y, z) 1 2 x, y, z (x, y, z) p. 13

z ± x z ± y p 2a 2, 2b 2. a b p = 2. p = 2 a 2 b 2 a 2 b 2 = (a + b)(a b). a b. 2 (x, y, z) 1 2 x, y, z 1 3.3 7 7 [00 ] (a) (b) (c) (d) (1) n n 6 15 n 30 1 < = n < = 300 6 15 n (2) m, n m + n, mn 12 m, n 6 1 < = m < = n < = 100 m + n, mn 12 m, n (3) l, m, n l + m + n, lm + mn + nl, lmn 5 l, m, n 5 (4) l, m, n l + m + n, lm + mn + nl, lmn 30 l, m, n 30 1 < = l < = m < = n < = 100 l + m + n, lm + mn + nl, lmn 30 l, m, n (a) (b) (c) (d) 8 8 99 a, b, c a 2 + b 2 = c 2 a, b (1) a b c (2) a a + c = 2d 2 d 14

9 9 a, b, c 06 3a = b 3, 5a = c 2 d 6 a d d = 1 (1) a 3 5 (2) a 3 5 (3) a 10 10 09 (1) 5 n 6n + 1 6n 1 (2) N 6N 1 6n 1 (n ) (3) 6n 1 (n ) 15

4 4.1 x y x, y, z 2 2x + 3y = 1, xy 2x 3y + 1 = 0, 1 x + 1 y + 1 z = 1, x2 3y 2 = 1 5x + 2y = 1, 7x 5y = 3 5x + 2y = 1, 5( 2) + 2(3) = 1, 5(1) + 2( 2) = 1, 5(3) + 2( 7) = 1, 5x + 2y = 1 37x + 13y = 1 4.2 6 {0} A a, b A a b A A d A A d A = { dn n : } 0 = a a A a A a = 0 a A n na A (n + 1)a = na ( a) A 16

a A na A na A n na A A d A a d a = dq + r 0 < = r < d d A dq A r = a dq A r > 0 d r = 0 A a d A { dn n : } d A n nd A A { dn n : } A = { dn n : } ( ) 1 7 ax + by = 1 (1) a b (2) a b (1) m, n a b d a = da, b = db am + bn = d(a m + b n) = 1 d 1 d = 1 a b (2) A A = { px + qy x, y Z } a = px 1 + qy 1 b = px 2 + qy 2 A a b = p(x 1 x 2 ) + q(y 1 y 2 ) A A. 6 A A d d A d = px 0 + qy 0 p = p 1 + q 0 A, q = p 0 + q 1 A p q d d p q p q d = 1 px + qy = 1 (x 0, y 0 ) ( ) 4.3 (i) m n m > = nq + 1 q + 1 17

(ii) n n 7(2) 14 14 (3) (4) 7(2) i = 0,, b 1 ai b r i. r i 0 b 1 0 < = i, j < = b 1 r i r j r i = r j. ai = bq i + r i aj = bq j + r j a(i j) = b(q i q j ) a b i j b. 0 (b 1) < = i j < = b 1 0 b i j = 0., i j = r i r j 0 < = i < = b 1 b i r i r i (0 < = i < = b 1) 0 b 1 k b q s r i = s i ai = bq i + s i q i s = k bq ai + b(q q i ) = k (x, y) = (i, q q i ) ax + by = k. ( ) 1 14(2) 8 a b ax + by = 1 1 x 0, y 0 (x, y) = (x 0 + bt, y 0 at), (t ) x y ax + by = 1 ax 0 + by 0 = 1 a(x x 0 ) + b(y y 0 ) = 0 a b x x 0 b bt x = x 0 + bt, y = y 0 at t x y ( ) ax + by = 1 18

4.4 3x + 2y = 1 127x + 52y = 1 a b 1 k a b k a b a b ax + by = 1 x y k ax + by = k a b ax + by = 1 (1) a > b a = bq + r, (0 < = r < b) (2) ax+by = (bq +r)x+by = rx+b(qx+y) y = qx+y ax+by = 1 rx + by = 1 (3) rx+by = 1 (x 0, y 0) y 0 = y 0 qx 0 (x 0, y 0 ) ax+by = 1 a b b r 1 sx + y = 1 x + ty = 1 (0, 1) (1, 0) ax + by = 1 127x + 52y = 1 (1) 127x + 52y = 1 (2) 127 = 52 2 + 23, y = 2x + y, 23x + 52y = 1 (3) 52 = 23 2 + 6, x = x + 2y, 23x + 6y = 1 (4) 23 = 6 3 + 5, y = 3x + y, 5x + 6y = 1 (5) 6 = 5 1 + 1, x = x + y, 5x + y = 1 (1) (x, y ) = (0, 1) (2) x = x + y (x, y ) = ( 1, 1) (3) y = 3x + y (x, y ) = ( 1, 4) (4) x = x + 2y (x, y ) = ( 9, 4) (5) y = 2x + y (x, y) = ( 9, 22), 127 ( 9) + 52 22 = 1143 + 1144 = 1 19

1 a b 5x + y = 1 1 0 1 4.5 11 11 80 n n > = 3 S = {a 1, a 2,, a n } S a i a j a i a j a j a i S (1) 2 a i > = 0 (i = 1, 2,, n) a i < = 0 (i = 1, 2,, n) (2) a 1, a 2,, a n 12 12 85 G (i),(ii) (i) m, n G m + n G (ii) m, n G m > n m n G G d G = {kd k } 13 13 99 3 2 (3) 0 x, C(x) x 3 C(12578) = 578, C(6) = 6 n 2 5 (1) x y 0 C(nx) = C(ny) C(x) = C(y) (2) C(nx) = 1 0 x (3) C(397x) = 1 0 x 14 14 00 a, b (1) 4m + 6n = 7 m, n (2) 3m + 5n = 2 (m, n) (3) k ak b r(k) k, l b 1 k l r(k) r(l) 20

(4) am + bn = 1 m, n 15 15 00 xy x y a, k a > = 2 L : ax + (a 2 + 1)y = k (1) L (2) k = a(a 2 + 1) x > 0, y > 0 L (3) k > a(a 2 + 1) x > 0, y > 0 L 16 16 08 (1) 3x + 2y < = 2008 0 (x, y) (2) x 2 + y 3 + z 6 < = 10 0 (x, y, z) 17 17 12 f(m, n) = m 2 mn + n 2 k (m, n) X(k) = {(m, n) m, n f(m, n) = k } (1) X(k) X(1) (2) k = 2, 4 X(k) (3) r X(2 r ) 21

5 5.1 a b m, a b m a b (mod. m) 9 2 a b m a b m a b m a b = mq a b m q 1, q 2 r 1, r 2 2 a b = mq a = mq 1 + r 1, b = mq 2 + r 2 mq = (q 1 q 2 )m + r 1 r 2 m q q 1 + q 2 = r 1 r 2 q q 1 + q 2 0 m q q 1 + q 2 > = m m r 1 r 2 < m q q 1 + q 2 = 0 r 1 r 2 = 0 a b m a = mq 1 + r, b = mq 2 + r a b = m(q 1 q 2 ) a b m ( ) 5.2 Z m Z : a a (mod. m) : a b (mod. m) b a (mod. m) : a b (mod. m) b c (mod. m) a c (mod. m) 22

m m, m m m m a k mk + a m m ( ) {0, 1, 2, 3, 4, 5, 6} {0, 1, 2, 3, 3, 2, 1} {7, 6, 9, 4, 10, 9, 13} 7 5.3 a b (mod. m) 10 a a (mod. m), b b (mod. m) a ± b a ± b (mod. m), ab a b (mod. m) (1) a a (mod. m), b b (mod. m), c c (mod. m), f(x, y, z, ) x, y, z, f(a, b, c, ) f(a, b, c, ) (mod. m) (2) a a b b m (a+b) (a +b ) = (a a )+(b b ) m, ab a b = (a a )b + a (b b ) m (1) (1) a a (mod. m) 1) N Na Na (mod. m) 2) α a α a α (mod. m) (1) Na α b β c γ Na α b β c γ (mod. m) (1) Na α b β c γ Na α b β c γ (mod. m) (2) ( ) ab = ac a 0 b = c 11 ac bc (mod. m) (c, m) = 1 a b (mod. m) 23

ac bc m N ac bc = (a b)c = mn m c a b m a b (mod. m) ( ) 12 a m ax 1 (mod. m) x a m ax + my = 1 (x, y) = (α, β) aα = 1 mβ 1 (mod. m) x = α ( ) 2 10 2 (1) 5.4 18 18 01 n n 9 n 3 9 19 19 03 5 x y z n x 2 = 7 2n (y 2 + 10z 2 ) (1) 2 3 0 1 (2) yz 3 (3) y z x n 20 20 [82 ] n f(x) = x n + a 1 x n 1 + + a n 1 x + a n (n > 1) (1) α f(x) = 0 α (2) k(> 1) k f(1), f(2),, f(k) k f(x) = 0 24

6 6.1 1, 2,, n n x φ(n) φ(1) = 1, (x = 1) φ(2) = 1, (x = 1) φ(3) = 2, (x = 1, 2) φ(4) = 2, (x = 1, 3) φ(5) = 4, (x = 1, 2, 3, 4) φ(6) = 2, (x = 1, 5) φ(n) p φ(p) = p 1 21 (2) p q φ(pq) = φ(p)φ(q) 13 a b φ(ab) = φ(a)φ(b) m = φ(a) α 1, α 2,, α m 1 a a n = φ(b) β 1, β 2,, β n α i, β j mn = φ(a)φ(b) a b ak + α i = bl + β j k l α i a a β j b b ab ab γ ij γ ij γ ij α i (mod. a), γ ij β j (mod. b) 1 ab i i γ i j γ ij ak + α i (mod. a) γ i j ak + α i (mod. b) γ ij γ i j a ab j ab mn φ(a)φ(b) < = φ(ab) (γ, ab) = 1 γ a b a α i, b β j i, j γ γ ab a b γ α i, β j φ(a)φ(b) > = φ(ab) ( ) 25

3 a = 14, b = 15 φ(14) = φ(2)φ(7) = 6, φ(15) = φ(3)φ(5) = 8 α i, β j α 1 = 1, α 2 = 3, α 3 = 5, α 4 = 9, α 5 = 11, α 6 = 13 β 1 = 1, β 2 = 2, β 3 = 4, β 4 = 7, β 5 = 8, β 6 = 11, β 7 = 13, β 8 = 14 α 4 = 9, β 6 = 11 14k + 9 = 15l + 11 k l k = 2, l = 2 19 γ 46 = 210 19 = 191 191 9 (mod. 14), 191 11 (mod. 15) 191 ab = 210 α 6 = 13, β 5 = 8 14k + 13 = 15l + 8 k l k = 5, l = 5 83 γ 65 = 83 83 13 (mod. 14), 83 8 (mod. 15) ab = 210 191 83 191 83 = 108 ab ab 6 8 = 48 γ ij 1 210 210 6.2 14 n a n a φ(n) 1 (mod. n) α 1, α 2,, α m, m = φ(n) n n a n aα i, aα j n α i, α j aα 1, aα 2,, aα m n n m aα 1, aα 2,, aα m α 1, α 2,, α m 1 n n aα 1 aα 2 aα m = a m α 1 α m α 1 α m (mod. n) α 1 α m n a m 1 (mod. n) 26

n φ(p) = p 1 p a ( ) a p 1 1 (mod. p) a p a a p a (mod. p) 6.3 21 21 03 (3) n m < = n m n 1 m f(n) (1) f(15) (2) p, q f(pq) (3) p e f(p e ) 22 22 06 2 N ϕ(n) N N ϕ(n) = {n n 1 < = n < N gcd(n, n) = 1 } gcd(a, b) a b A A ϕ(6) = {1, 5} = 2, ϕ(15) = {1, 2, 4, 7, 8, 11, 13, 14} = 8 (1) p q N = pq (i) N n gcd(n, n) 1 (ii) ϕ(n) (2) p q N = pq N ϕ(n) p q N ϕ(n) (3) N = 84773093 ϕ(n) = 84754668 N = pq (p > q) p q ( p q ), 320 2 = 102400 322 2 = 103684 324 2 = 104976 326 2 = 106276 328 2 = 107584 330 2 = 108900 27

23 23 95 n f(n), g(n) f(n) = n 7 ( 7 ) g(n) = 3f k n. (1) n f(n 7 ) = f(n). (2) n g(n). g(n) (2) 24 24 p n p 1 p 1 A (1) k A nk p r k {r k k A} A (2) n p 1 1 p k=1 25 25 (1) p 1 < = r < = p 1 r, r p C r = p p 1 C r 1, p C r p. (2) p 2 p p. (3) n n p p,. 26 26 10 p r. (1) x 1, x 2,, x r (x 1 + x 2 + + x r ) p p x 1 p2 p 1 x 2 x r r p 1, p 2,, p r 0 p 1 + p 2 + + p r = p. (2) x 1, x 2,, x r (x 1 + x 2 + + x r ) p (x p 1 + x p 2 + + x p r ) p (3) r p r p 1 1 p 28

7 7.1 R[x] Q[x] R[x] Q[x] C[x] K Q R C K K[x] x deg f(x) f(x) 15 f(x), g(x) (deg g(x) > = 1) f(x) = g(x) q(x) + r(x), 0 < = deg r(x) < deg g(x) q(x), r(x) deg f(x) = n, deg g(x) = m q(x) r(x) n n < m q(x) = 0, r(x) = f(x) 0 n 1 f(x) g(x) n, m ax n, bx m f 1 (x) = f(x) a b xn m g(x), deg f 1 (x) < deg f(x) f 1 (x) = g(x) q 1 (x) + r 1 (x), 0 < = deg r 1 (x) < deg g(x) q 1 (x), r 1 (x) f(x) = g(x) a b xn m + f 1 (x) = g(x) a b xn m + g(x) q 1 (x) + r 1 (x) { } a = g(x) b xn m + q 1 (x) + r 1 (x) q(x) = a b xn m + q 1 (x) r(x) = r 1 (x) 2 f(x) = g(x) q 1 (x) + r 1 (x) = g(x) q 2 (x) + r 2 (x) g(x) {q 1 (x) q 2 (x)} = r 2 (x) r 1 (x) q 1 (x) q 2 (x) 0 deg(r 2 (x) r 1 (x)) > = deg g(x) deg r 1 (x) < deg g(x), deg r 2 (x) < deg g(x) deg(r 2 (x) r 1 (x)) < deg g(x) q 1 (x) = q 2 (x) r 1 (x) = r 2 (x) ( ) 29

7.2 f(x) g(x) f(x) = q(x)g(x) q(x) f(x) g(x) f(x) g(x) f(x) g(x) f(x) g(x) 0 { } 1 x 2 + 3x + 2 = (x + 1)(x + 2) = {3(x + 1)} (x + 2) = 3 K[x] 0 0 K[x] 0 0 f(x) (0 ) f(x) x + 1, 3(x + 1), 2(x + 1) f(x) = x 4 4 Q[x] f(x) = (x 2 2)(x 2 + 2) x 2 2, x 2 + 2 R[x] f(x) = (x 2)(x + 2)(x 2 + 2) x 2, x + 2, x 2 + 2 C[x] f(x) = (x 2)(x + 2)(x 2i)(x + 2i) 7.3 27 27 90 f(x) g(x) f(x)p(x) = g(x) p(x) d(x) 2 f(x) g(x) d(x) f(x) g(x) 30

d(x) 2 f(x) g(x) d(x) f(x) g(x) (f(x) = g(x) = 0 ), (1) f(x) f(x) (2) 0 f(x) g(x) f(x) f(x) g(x) (3) f(x) 0 g(x) f(x) r(x) d(x) r(x), f(x) d(x) f(x) g(x) 28 28 06 1 3 Q(x) 2 P (x) Q(x) {P (x)} 2 Q(x) 2 Q(x) = 0 29 29 06 1 3 1 A(x) B(x) C(x) {A(x)} 2 + {B(x)} 2 = {C(x)} 2 A(x) B(x) C(x) 30 30 02 f(x) = x 1, g(x) = (x + 1) 3 p(x)f(x) + q(x)g(x) = 1 p(x), q(x) p(x) p(x) 1 31

8 2 1 1 (1) a, b 3 a b 3 q 1, q 2 r 1, r 2 r 1, r 2 = 1, 2 a = 3q 1 + r 1, b = 3q 2 + r 2 ab = 3(3q 1 q 2 + q 1 r 2 + q 2 r 1 ) + r 1 r 2 r 1 r 2 = 1, 2, 4 ab 3 ab 3 a b 3 (2) (1) a b 3 a 3 a + b 3 a 3 b 3 b 3 a b 3 (3) a 2 + b 2 = (a + b) 2 2ab a + b 3 2ab 3 2 3 ab 3 (1) a b 3 (0) mn p mn p m n p mn p 3 4 m n p (1) ab p a b p a p a = pa a + b p pn b = (a + b) a = pn pa b p b p a p a b p (2) a 2 + b 2 = (a + b) 2 2ab a + b p 2ab p p 2 ab p (1) a b p (3) a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a + b a 2 ab + b 2 p a + b p a 2 + b 2 p (2) a b p a 2 ab + b 2 p a 2 + b 2 p ab p a p a 2 + b 2 p b 2 p p b p b p a b p 32

a 3 + b 3 = (a + b)(a 2 + b 2 ) ab(a + b) ab(a + b) p a + b p 2 a, b p ab p a 2 b 2 p 1 a 2 b 2 p p a b p (4) a p a n + b n p b n p p b p a b p p a b p n > = 2 a n+1 + b n+1 = (a + b)(a n + b n ) ab(a n 1 + b n 1 ) a n 1 + b n 1 p a + b a 2 + b 2 p 2 a, b p a b p a b p 2 2 (1) 0 < = r(n) < = 7 0 < a r(a) < 4 3 r(b) < = 28 3 a r(a) 8 a r(a) = 8 8 < 4 r(b) 6 < r(b) 3 r(b) = 7 (2) b r(b) = 8, r(ab) = 7 b = 8 + r(b) = 15 ab = 8k + 7 b = 15 ab = 8a + 7a 8(k a) = 7(a 1) 8 7 a 1 8 r(a) = 1 a = 8 + r(a) = 9 3 3 (1) k k + 1 d k = dk, k + 1 = dh k, h 1 = dh k = d(h k ) d = 1 k k + 1. 33

(2) 2k + 1 k + 1 1 2k + 1 (k + 1) = k k k k + 1 2k + 1 k + 1 2k +1 k 2k +1 2 k = 1 1 2k +1 = n 2, k +1, k n k + 1 k 1 n 2 k + 1 k 1 2k+1 = n 2, k+1, k n, k, k+1. (3) n 2 = 2k + 1 n 2 + k 2 = (k + 1) 2 3 n, k, k + 1 3 (2) n 3 3 1 n = 11 n 2 = 121 = 2 60 + 1 (11, 60, 61) 11 2 = 121, 60 2 = 3600, 61 2 = 3721 4 4 (1) b(p 2 + q 2 ) = apq 1 b a b pq b (2) p q g p = gp, q = gq, (p q ) 1 bg 2 (p 2 + q 2 ) = ag 2 p q b(p 2 + q 2 ) = ap q 2 (1) p q b p q = bk 2 p 2 + q 2 = ak 34

k 1 p 2 + q 2 p q p 2 + q 2 + 2p q p q (p + q ) 2 p q p + q p q p q p + q p q k = 1 b = p q a = p 2 + q 2 a + 2b = p + q 5 5 (1) n + 2 n 2 + 1 d (n 2 + 1) (n + 2)(n 2) = 5 d 5 d 5 d = 1, 5 (2) (1) n + 2 n 2 + 1 1 5 n + 2 5 (1) n 2 + 1 5 n + 2 5 n n = 5k 2 (k = 1, 2, ) (3) 4(n 2 + 1) (2n + 1)(2n 1) = 5 2n + 1 n 2 + 1 1 5 2n + 1 5 2n + 1 5 n 2n + 1 = 5(2k 1) (k = 1, 2, ) n n = 5k 3 (k = 1, 2, ) 6 6 (1) a 1 = 1998, b 1 = 185 1998 = 185 10 + 148 a 2 = 185, b 2 = 148 185 = 148 1 + 37 a 3 = 148, b 3 = 37 148 = 37 4 + 0 a 4 = 37, b 4 = 0 a 5 = 37, b 5 = 0 35

(2) b n 0 b n+1 a n b n 0 < = b n+1 < b n b n = 0 {b n } b n+1 = b n k, l, n b n > = b n+1 ( b n = 0 ) (3) b n = 0 n b n b n > b n+1 {b n n = 1, 2, } b n = 0 n (4) (a, b) a b b k 0 (a k, b k ) = (b k, b k+1 ) a k b k q k a k = b k q k + b k+1 a k b k b k+1 b k b k+1 b k b k+1 (b k, b k+1 ) (a k, b k ) < = (b k, b k+1 ) b k b k+1 a k (a k, b k ) > = (b k, b k+1 ) (a k, b k ) = (b k, b k+1 ) (a k, a k+1 ) = (a k+1, a k+2 ) (3) N b N 1 0, b N = 0 (k, l) = (a 1, a 2 ) = = (a N 1, a N ) = (a N, b N ) = a N N < = n n 3 7 7 (1) n 30 6 15 (b) 6 50 15 20 6 15 30 10 50 + 20 10 = 60 (2) m + n = 12k, mn = 12l mn = m(12k m) = 12l m 2 = 12(mk l) m 2 12 m 2 3 6 n = 12 m n 6 m = 6, n = 12 m + n = 18 12 (c) 36

m = 6k, n = 6l m + n = 6(k + l), mn = 36kl m + n, mn 12 k l k = 1 l = 1, 3,, 15 8 k = 2 l = 2, 4,, 16 8 k = 3 l = 3,, 15 7 k = 4 l = 4,, 16 7 k = 15 15 1 k = 16 16 1 2(8 + 7 + + 1) = 72 (3) l + m + n, lm + mn + nl, lmn 5 lmn 5 l 5 m + n = l, mn = l(m + n) 5 m, n 5 m 5 n = m 5 l, m, n l, m, n 5 (a). (4) (3) l + m + n, lm + mn + nl lmn 5 30 = 5 6 5 6 6 l + m + n, lm + mn + nl, lmn 6 lmn 6 l 2 (3) m + n, mn 2 l, m, n 2 l 3 m + n, mn 3 l, m, n 3 l, m, n 6 l, m, n 30 (a) 30 (l, m, n) = (30, 30, 30), (30, 30, 60), (30, 30, 90), (30, 60, 60), (30, 60, 90), (30, 90, 90) (60, 60, 60), (60, 60, 90), (60, 90, 90) (90, 90, 90) 10 8 8 (1) a b c a = 2k + 1 b = 2l + 1, c = 2m a 2 + b 2 = 4(k 2 + k + l 2 + l) + 2, c 2 = 4m 2 37

4 a 2 + b 2 = c 2 b c (2) a 2 + b 2 = c 2 b 2 = (c a)(c + a) (1) c a, c + a c + a 2 c a 2 c + a 2 > c a 2 ( ) 2 b = c a c + a 2 2 2 > = 1 p b p > 1 2 p c a 2 = kp, c + a 2 = lp c = (k + l)p, a = (l k)p p a b a b p > 1 p c a 2 c + a 2 ( ) 2 b c a c + a 2 2 2 d 9 9 a + c 2 (1) b 3 3 3 b 3 b = 3b 3a = b 3 = d 2 3a = (3b ) 3 = 27b 3 a = 9b 3 a 3 c 5 c = 5c 5a = c 2 5a = 5c 2 a = 5c 2 a 5 (2) a 3 5 p a = pa p(3a ) = b 3, p(5a ) = c 2 (1) b c p b = p l b, c = p m c b, c p 3a = p 3l b 3, 5a = p 2m c 2 p 3, 5 a p 1 3 2 2 a p 6 d 6 a d d = 1 a 3 5 38

(3) a = 3 e 5 f 1 < = e, f < = 5 3a = 3 e+1 5 f = b 3, 5a = 3 e 5 f+1 = c 2 e + 1, f 3 e, f + 1 2 1 < = e, f < = 5 e = 2, f = 3 a = 3 2 5 3 = 1125 10 10 (1) 5 p 3 6 1 5 p 5 n 6n + 1 6n 1 (2) N 6N 1 > = 5 6N 1 3 3 6n + 1 6n 1 6n + 1 6n 1 + 1,, 6n l + 1 6 1 6u + 1, 6v + 1 (6u + 1)(6v + 1) = 6(6uv + u + v) + 1 6 1 6n 1 + 1,, 6n l + 1 (6n 1 + 1) e1 (6n l + 1) e l 1 6 1 6N 1 6 1 5 6N 1 6n 1 (n ) (3) 6n 1 (n ) k p 1,, p k L = 6p 1 p k 1 L p 1,, p k 1 L 6N 1 (2) 6n 1 p 1,, p k 6n 1 (n ) k 4 11 11 (1) S M m m < 0 < M M m m M S M < M m m M < m M m 39

(2) S d S d a qd < = a < (q + 1)d q r = a qd r > 0 a d S 1 < = j < q a jd S a jd d = a (j +1)d S r = a dq S d S r = 0 S d S md 1 < j < = m jd S jd d = (j 1)d S md, (m 1)d,, d S 0 S m = n m = n 1 S = {nd, (n 1)d,, d}, {(n 1)d, (n 2)d,, d, 0} a 1, a 2,, a n d 0 d 12 12 j jd G d + jd = (j + 1)d G {kd k } G G a a d q r a = dq + r dq G r = a dq S r > 0 d G r = 0 G d G {kd k } 13 13 (1) x > = y C(nx) = C(ny) nx ny 1000 nx ny = n(x y) n 1000 x y 1000 C(x) = C(y) 40

(2) (1) C(0), C(n), C(2n),, C(999n) {C(0), C(n), C(2n),, C(999n)} 1000 {0, 1,, 999} 1000 {C(0), C(n), C(2n),, C(999n)} = {0, 1,, 999} C(0),, C(999n) 1 C(nx) = 1 0 x (3) 379x = 1 + 1000y (x, y) 1 = 397x 1000y = 397x (397 2 + 206)y = 397(x 2y) 206y = (206 + 191)(x 2y) 206y = 206(x 3y) + 191(x 2y) = (191 + 15)(x 3y) + 191(x 2y) = 191(2x 5y) + 15(x 3y) = (15 12 + 11)(2x 5y) + 15(x 3y) = 15(25x 63y) + 11(2x 5y) 15 3 + 11 ( 4) = 1 { 25x 63y = 3 2x 5y = 4 x = 267, y = 106 397( 267) 1000( 106) = 1 (m, n) 397m 1000n = 1 397(m + 267) 1000(n + 106) = 0 397 1000 m + 267 1000 m + 267 = 1000t n + 106 = 397t (x, y) t (x, y) = ( 267 + 1000t, 106 + 397t) x t = 1 C(397x) = 1 0 x x = 1000 267 = 733 C(733 397) = C(291001) = 1 4 (2) (3) 14 14 (1) 4m + 6n = 7 m, n 2 2 1 m, n 41

(2) 3m + 5n = 2 (m, n) ( 1, 1) (m, n) 3m + 5n = 2 3( 1) + 5(1) = 2 3(m + 1) + 5(n 1) = 0 3 5 m + 1 5 m + 1 = 5t n 1 = 3t (m, n) = ( 1 + 5t, 1 3t) (m, n) = ( 1 + 5t, 1 3t) (t ) (3) r(k) = r(l) ak al b (a b ) k l b 1 < = k, l < = b 1 (b 2) < = k l < = b 2 k l = 0 k l r(k) r(l) (4) A = {1, 2,, b 1} B = {r(k) k = 1, 2,, b 1} k r(k) = 0 ak b a b k b k = 1, 2,, b 1 r(k) b 0 B A (3) k l r(k) r(l) k = 1, 2,, b 1 r(k) B b 1 A A = B B r(k) = 1 42

ak 1 b ak 1 = bl (k, l) (m, n) = (k, l) 15 15 (1) a( ak) + (a 2 + 1)k = k 1 ( ak, k) L (2) (m, n) L am + (a 2 + 1)n = k 2 2 1 a(m + ak) + (a 2 + 1)(n k) = 0 3 a a 2 a 2 + 1 a a 2 + 1 3 m + ak a 2 + 1 t m + ak = (a 2 + 1)t n k = at L t { m = ak + (a 2 + 1)t n = k at L { m = ak + (a 2 + 1)t > 0 n = k at > 0 t k a > t > ak a 2 + 1 4 k = a(a 2 + 1) 4 a 2 + 1 > t > a 2 t L (3) k > a(a 2 + 1) 4 k a ak a 2 + 1 = k a(a 2 + 1) > 1 t L 43

16 16 (1) k 3x + 2y = k 0 (x, y) k = 0, 1,, 2008 k 3x + 2y = k (x, y) 3k + 2( k) = k 3(x k) + 2(y + k) = 0 2 3 x k 2 y + k 3 (x, y) t x = k 2t, y = k + 3t x > = 0, y > = 0 k 3 < = t k < = 2 k t 3x + 2y = k 0 (x, y) m k t t k = 6m 2m < = t < = 3m m + 1 k = 6m + 1 2m + 1 3 < = t < = 3m + 1 2 m k = 6m + 2 2m + 2 3 < = t < = 3m + 1 m + 1 k = 6m + 3 2m + 1 < = t < = 3m + 3 2 m + 1 k = 6m + 4 2m + 4 3 < = t < = 3m + 2 m + 1 k = 6m + 5 2m + 5 3 < = t < = 3m + 5 2 m + 1 m 6m + 5 2008 = 6 334 + 4 m = 0, 1,, 334 6m + 5 2009 335 334 (6m + 5) 335 = 6 167 335 + 5 335 335 m=0 = (1002 + 4) 335 = 337010 ( ) (2) 3x + 2y + z < = 60 3x + 2y = k (x, y) z 0 < = z < = 60 k 60 k + 1 (x, y) z 3x + 2y = k 3x + 2y + z < = 60 (x, y, z) k = 6m k = 6m + 1 k = 6m + 2 k = 6m + 3 k = 6m + 4 k = 6m + 5 (m + 1)(60 6m + 1) m(60 6m) (m + 1)(60 6m 1) (m + 1)(60 6m 2) (m + 1)(60 6m 3) (m + 1)(60 6m 4) 44

(m + 1)(300 30m 9) + m(60 6m) = 36m 2 + 321m + 291 m = 0,, 9 3x + 2y = 60 ( z 0 ) 11 9 ( 36m 2 + 321m + 291) + 11 = 6 9 10 19 + 321 45 + 2910 + 11 m=0 = 10260 + 14445 + 2921 = 7106 ( ) 17 17 (1) f(m, n) = k m 2 mn + n 2 k = 0 m 2 m m 2 D n D = n 2 4(n 2 k) = 3n 2 + 4k > = 0 0 < = n 2 4k < = n n 3 m 2 X(k) k = 1 0 < = n 2 4 < = n = 0, ±1 m 3 X(1) = {(1, 0), ( 1, 0), (0, 1), (0, 1), (1, 1), ( 1, 1)} (2) f(m, n) = m 2 mn + n 2 = k k m n f(m, n) m n f(m, n) k = 2 0 < = n 2 < = 8 3 n = 0 m2 = 2 X(2) 0 k = 4 m n m = 2p n = 2q m 2 mn + n 2 = 4p 2 4pq + 4q 2 = 4 p 2 pq + q 2 = 1 (1) (p, q) 6 (2p, 2q) f(m, n) = 4 X(4) 6 (3) (2) f(m, n) = 2 r (r ) m = 2p n = 2q f(m, n) = m 2 mn + n 2 = 2 r f(2p, 2q) = 4p 2 4pq + 4q 2 = 2 r f(p, q) = p 2 pq + q 2 = 2 r 2 X(2 r ) X(2 r 2 ) (2) X(2) 0 X(2 2 ) 6 X(2 r ) r 0 6 45

1 r X(2 r ) 0 n m D k = 2 r D = n 2 4(n 2 k) = 3n 2 + 2 r+2 = N 2 1 1 3 N 2 3 N 3 0 N = 3s ± 1 1 2 2 2l = (3 + 1) l 1 2 r 1 n N y 2 x 2 xy+y 2 = x 2 xy + y 2 = 2 2l 2 2l 2 l π 4 6 2 l O 2 l 2 l x 5 18 18 1 n 9 n 3 = n 2 (n 7 n) n 3 9 3 n = 3k ± 1 n 9 n 3 = (3k ± 1) 3 {(3k ± 1) 6 1} = (27k 3 ± 27k 2 + 9k ± 1){(3k ± 1) 6 1} (±1){(3k ± 1) 6 1} (mod. 9) = (±1){(9k 2 ± 6k + 1) 3 1} (±1){(36k 2 ± 12k + 1)(±6k + 1) 1} (mod. 9) (±1){(±3k + 1)(±6k + 1) 1} (mod. 9) = (±1)(18k 2 ± 9k + 1 1) 0 (mod. 9) n 9 n 3 9 2 n 9 n 3 = n 3 (n 3 1)(n 3 + 1) = (n 1)n(n + 1)n 2 (n 2 + n + 1)(n 2 n + 1) = (n 1){(n 1) 2 + 3n} n 3 (n + 1){(n + 1) 2 3n} n 3 n 3 9 n 3 1 (n 1){(n 1) 2 + 3n} 9 n 3 2 (n + 1){(n + 1) 2 + 3n} 9 3 46

19 19 (1) n 3 q r r = 0, 1, 2 n 2 = (3q + r) 2 = 3(3q 2 + 2qr) + r 2 n 2 r 2 3 r 2 = 0, 1, 4 = 3 + 1 2 3 0 1 n 2 3 0 n 3 (2) y z 3 (1) y 2 = 3k + 1, z 2 = 3l + 1 2 7 2n = (6 + 1) 2n = 2n 1 i=0 2nC i 6 2n i + 1 7 2n = 3m + 1 k, l, m 7 2n (y 2 + 10z 2 ) = (3m + 1){3k + 1 + 10(3l + 1)} 7 2n (y 2 + 10z 2 ) 3 2 = (3m + 1){(3(10l + k + 3) + 2} x 2 3 0 1 3 y z 3 yz 3 (3) yz 3 y z y = 3 z = 3 y = 3 x 2 = 7 2n (9 + 10z 2 ) 7 2n 9 + 10z 2 N 2 10z 2 = (N + 3)(N 3) N N + 3 = N 3 + 6 N + 3 N 3 (N + 3, N 3) = (5z 2, 2), (z 2, 10), (10z, z), (5z, 2z), (10, z 2 ) z 2 z = 2 z = 3 y 2 + 90 y 2 + 90 = N 2 (N + y)(n y) = 90 N + y = N y + 2y N + y N y 4 90 y y = 3, z = 2 x 2 = 7 2n (9 + 40) = 7 2n+2 x = 7 n+1 47

20 20 (1) α α = q p q p f ( ) q = p ( q p ) n ( ) n 1 q + a 1 + + a n 1 p ( ) q + a n = 0 p q n = p ( a 1 q n 1 + + a n 1 qp n 2 + a n p n 1) p p q n p = ±1 α (2) f(x) = 0 α (1) α α k q r j α j r j = (α r)(α j 1 + α j 2 r + + r j 1 ) α r = kq α j r j k α j = r j + N j k a 0 = 1, N 0 = 0 f(α) = n n a j α n j = a j (r n j + N n j k) = j=0 n n a j r n j + k a j N n j = 0 j=0 j=0 j=0 n n a j r n j = f(r) = k a j N n j j=0 j=0 k k f(0), f(1), f(2),, f(k 1) n n 1 f(r) k f(0) = a n f(k) = a j k n j = k a j k n j 1 + a n f(0) k f(k) k k j=0 f(1), f(2),, f(k) k (2) j=0 5 10 α r (mod. k) f(α) f(r) (mod. k) 6 1 1 f(x) = x + a a (1) 6 21 21 48

(1) 15 15 1, 2, 4, 7, 8, 11, 13, 14 f(15) = 8 (2) 1 < = m < = pq pq p q {p, 2p,, pq}, {q, 2q,, pq} pq q + p 1 f(pq) = pq (q + p 1) = (p 1)(q 1) (3) 1 p e p e p p e 1 1 p, 2 p,, p e 1 p ( f(p e ) = p e p e 1 = p e 1 1 ) p 22 22 (1) (i) gcd(n, n) 1 p q N n p, 2p, 3p,, (q 1)p q, 2q, 3q,, (p 1)q (ii) 1 < = n < N gcd(n, n) 1 (i) (q 1) + (p 1) gcd(n, n) = 1 ϕ(n) = pq 1 (q 1) (p 1) = (p 1)(q 1) (2) N = pq (ii) ϕ(n) = N (p + q) + 1 p + q = N + 1 ϕ(n), pq = N p q x x 2 {N + 1 ϕ(n)}x + N = 0 49

(3) N + 1 ϕ(n) = 18426 = 2 9213 p, q = 9213 ± 9213 2 84754668 = 9213 ± 106276 = 9539, 8887 23 23 (1) m, n 7 m, n, i, j, m = 7m + i, n = 7n + j mn = (7m + i)(7n + j) = 7(7m n + m j + n i) + ij, f(mn) = f(ij).,, n 7, n 2, n 3,, n 7 7,. n 0 1 2 3 4 5 6 n 2 0 1 4 2 2 4 1 n 3 0 1 1 6 1 6 6 n 4 0 1 2 4 4 2 1 n 5 0 1 4 5 2 3 6 n 6 0 1 1 1 1 1 1 n 7 0 1 2 3 4 5 6, n f(n 7 ) = f(n) 7 7 n 7 n 7 n (2) (1), k, k 7 k 7 7 7 7 k n+6 k n = k n 1 (k 7 k) = 7l k=1 k=1 k=1 g(n + 6) = g(n) (l ), 1 < = n < = 6., (1) g(1) = 3f(1 + 2 + 3 + 4 + 5 + 6 + 0) = 3f(21) = 0 g(2) = 3f(1 + 4 + 2 + 2 + 4 + 1 + 0) = 3f(14) = 0 g(3) = 3f(1 + 1 + 6 + 1 + 6 + 6 + 0) = 3f(21) = 0 g(4) = 3f(1 + 2 + 4 + 4 + 2 + 1 + 0) = 3f(14) = 0 g(5) = 3f(1 + 4 + 5 + 2 + 3 + 6 + 0) = 3f(21) = 0 g(6) = 3f(1 + 1 + 1 + 1 + 1 + 1 + 0) = 3f(6) = 18 50

, n = 6 g(6) = 18 24 24 (1) {r k k A} B n p r k = 0 nk p k p 1 < = k < = p 1 0 B B A i, j A r i = r j n(i j) p i j p p + 2 < = i j < = p 2 i j = 0 i j r i r j B p 1 A B = A (2) k A q k nk = pq k + r k (ni)(nj) = p(pq i q j + q i r j + q j r i ) + r i r j n 2n (p 1)n = pn + r 1 r 2 r p 1 N B = A r 1 r 2 r p 1 = (p 1)! n p 1 (p 1)! (p 1)! = pn (p 1)! p n p 1 1 p 25 25 (1) (2) r p C r = rp! r!(p r)! = p(p 1)! (r 1)!{(p 1) (r 1)}! = p p 1C r 1 p, r p p C r p. p 1 2 p = (1 + 1) p = 1 + pc r + 1 (1) 2 p 2 p 2 (p > 2) 0(p = 2) (3) n p n p.,. (i) n = 1. n = 2 (2) r=1 51

(ii) n = k. k p k = pm M. p 1 p 1 (1 + k) p = 1 + pc r k r + k p = 1 + pc r k r + pm + k r=1 p 1, pc r k r p N pn. r=1 r=1 (1 + k) p = 1 + pn + k (k + 1) p (k + 1) = pn, n = k + 1. (iii), n, n p n p., n p p n p 26 26 (1) (x 1 +x 2 + +x r ) p (x 1 +x 2 + +x r ) p x 1, x 2,, x r p x 1 p2 p 1 x 2 x r r x 1, x 2,, x r p 1, p 2,, p r N N x 1, x 2,, x r p 1, p 2,, p r p 1, p 2,, p r x 1, x 2,, x r p 1!p 2! p r! p! p 1!p 2! p r!n = p! p x 1 p2 p 1 x 2 x r r N p! p 1!p 2! p r! 1 p p 1 + p 2 + + p r = p p 1, p 2,, p r 1 (2) p 1, p 2,, p r 0 p j p! p 1!p 2! p r! = p (p 1)! p j p 1! (p j 1)! p r! p j p! p 1!p 2! p r! = p (p 1)! p 1! (p j 1)! p r! (1) p p p j < p p! p p j p p 1!p 2! p r! 52

(x 1 + x 2 + + x r ) p (x 1 p + x 2 p + + x r p ) 2 p x 1 p2 p 1 x 2 x r r p j p j < p p x 1, x 2,, x r p (3) 2 x 1, x 2,, x r 1 (2) r p r = r(r p 1 1) p r p r p 1 1 p 7 27 27 (1) p(x) = 1 f(x) = f(x)p(x) f(x) f(x) (2) 0 f(x) g(x) (1) f(x) f(x) g(x) f(x) f(x) f(x) f(x) g(x) f(x) f(x) g(x) (3) g(x) f(x) q(x) g(x) = f(x)q(x) + r(x) d(x) r(x), f(x) d(x) g(x) d(x) f(x) g(x) f(x) g(x) D(x) d(x) < = D(x) r(x) = g(x) f(x)q(x) D(x) r(x) D(x) r(x), f(x) D(x) < = d(x) D(x) = d(x) d(x) f(x) g(x) 28 28 1 ( ) Q(x) 2 P (x) Q(x) 1 A(x) ax + b P (x) = Q(x)A(x) + ax + b {P (x)} 2 = {Q(x)A(x)} 2 +2(ax + b)q(x)a(x) + (ax + b) 2 53

{P (x)} 2 Q(x) {P (x)} 2 = Q(x)B(x) Q(x) [ B(x) Q(x){A(x)} 2 2(ax + b)a(x)] = (ax + b) 2 Q(x) 2 2 B(x) Q(x){A(x)} 2 2(ax + b)a(x) c P (x) Q(x) 0 c 0 Q(x) Q(x) = 1 (ax + b)2 c Q(x) 2 a 0 Q(x) = 0 x = b a 2 () Q(x) = 0 α β Q(x) = a(x α)(x β) α β x α x β f(x) f(x) Q(x) f(x) x α x β P (x) x α {P (x)} 2 x α {P (x)} 2 x α P (x) x α {P (x)} 2 Q(x) {P (x)} 2 x α x β P (x) x α x β P (x) Q(x) α = β 2 Q(x) = 0 x = α 29 29 1 ( ) A(x) B(x) C(x) a, p b, q A(x) = ac(x) + p B(x) = bc(x) + q 1 ab 0 {A(x)} 2 + {B(x)} 2 = {C(x)} 2 a 2 {C(x)} 2 + 2apC(x) + p 2 + b 2 {C(x)} 2 + 2bqC(x) + q 2 = {C(x)} 2 1 C(x) cx + d c 0 x 2 a 2 c 2 + b 2 c 2 = c 2 1 a 2 + b 2 = 1 2apC(x) + p 2 + 2bqC(x) + q 2 = 0 54

x 2apc + 2bqc = 0 ap + bq = 0 2 p 2 + q 2 = 0 3 a 0 2 3 q = 0 3 p = 0 2 p = bq a b 2 q 2 a 2 + q 2 = b2 + a 2 a 2 q 2 = 1 a 2 q2 = 0 A(x) B(x) C(x) () {A(x)} 2 + {B(x)} 2 = {C(x)} 2 {B(x)} 2 = {C(x) + A(x)}{C(x) A(x)} C(x) + A(x) C(x) A(x) 1 {B(x)} 2 k (k 0) C(x) + A(x) = kb(x) C(x) A(x) = 1 k B(x) A(x) = k2 1 2k B(x), C(x) = k2 + 1 2k B(x) A(x) C(x) 1 k 2 ± 1 0 A(x) = k2 1 k 2 C(x), B(x) = 2k + 1 k 2 + 1 C(x) A(x) B(x) C(x) 30 30 1 (x + 1) 3 x 1 (x + 1) 3 = (x 1)(x 2 + 4x + 7) + 8 ( 1 8 x2 + 1 2 x + 7 ) (x 1) + 1 8 8 (x + 1)2 = 1 p(x)f(x) + q(x)g(x) = 1 { (x 1) p(x) + 1 8 x2 + 1 2 x + 7 } { + (x + 1) 3 q(x) 1 } = 0 8 8 55

x 1 (x + 1) 3 T (x) p(x) + 1 8 x2 + 1 2 x + 7 8 = (x + 1)3 T (x) q(x) 1 = (x 1)T (x) 8 p(x) = 1 8 x2 1 2 x 7 8 + (x + 1)3 T (x) q(x) = 1, (x 1)T (x) 8 (T (x) ) T (x) = 0 p(x) = 1 8 x2 1 2 x 7 8 q(x) = 1 8 1 T (x) = 1 p(x) = x 3 + 23 8 x2 + 5 2 x + 1 8 q(x) = x + 7 8 2 x = 1, x = 1 q(1)(1 + 1) 3 = 1, p( 1)( 1 1) = 1 p(x), q(x) P (x), Q(x) p(x) = (x + 1)P (x) 1 2, q(x) = (x 1)Q(x) + 1 8 { (x 1) (x + 1)P (x) 1 } { + (x + 1) 3 (x 1)Q(x) + 1 } = 1 2 8 (x 2 1)P (x) + (x 2 1)(x + 1) 2 Q(x) = 1 (x + 1)3 8 + x 1 2 = (x2 1)( x 3) 8 P (x) = (x + 1) 2 Q(x) x + 3 8 Q(x) = 0 P (x) = x + 3 8 56

( p(x) = (x + 1) x + 3 ) 1 8 2 = 1 8 x2 1 2 x 7 8 q(x) = 1 8 p(x), q(x) (x 1)p(x) ( + (x + 1) 3 q(x) = 1 (x 1) 1 8 x2 1 2 x 7 ) ( ) 1 + (x + 1) 3 8 8 = 1 ( ) 57