Congruence Equations

Aº+

»ñ¬È/P
.ó.

Chapter 4

Congruence Equations

ÉQ3 modulo m ì “≡” |A “=” ×øºÕ, &Æ!øݝ|"DŠ]PÝ®Þ.

9øÝ]PµÌ congruence equation. ÍýL, &Æ©D¡ŠŽŽóÝ congruence equation. 9×a, &ÆÞ"DŠ congruence equation Ý×æJ, ¬D¡»yõ§

|CŠ×gÝ congruence equation.

4.1. Š Congruence Equation ÝæJ

›×J;ó94P f(x) (Ç f(x) = cnxⁿ+ · · · + c₁x + c₀,Í ci ∈ Z),ãy f(x) Ý;ó ÎJó, Þ x ‚×Jó a `, f(a) ) Jó. .hu› m ∈ N, &Ɲ|®§øÝJó aº¸ÿ f(a) ≡ 0 (mod m) (Ç m|f(a)). 09øXbÝJóŠµÎXÛ݊ congruence equation.

› f(x) = cnxⁿ+ · · · + c₁x + c₀,Í ci ∈ Z. uáEy m ∈ N, a ∈ Z Î f(x) ≡ 0 (mod m)Ý×͊, Ç f(a) ≡ 0 (mod m). ƒ' b ≡ a (mod m), ã Proposition 3.2.2 á, EŒ i ∈ N /b bⁱ ≡ aⁱ (mod m). ã!× Proposition á cibⁱ ≡ c_iaⁱ (mod m), ‚ ÿ f(b) ≡ f(a) (mod m). ôµÎ1, u x = a Î f(x) ≡ 0 (mod m) Ý×ÍJóŠ, JE

Œ b ∈ Z ”• b ≡ a (mod m), x = b ù f(x) ≡ 0 (mod m) Ý×͊. X|u x = a Î f (x) ≡ 0 (mod m)Ý×ÍJóŠ, &Æ;ðº1 x ≡ a (mod m) Î f(x) ≡ 0 (mod m) Ý

×͊. Qbb̀3 modulo m ìõ a !õÝJóºÎ f(x) ≡ 0 (mod m) ݊. &ÆÄ6.9°Šà modulo m Ý!õvÝ]PI¶ì, 9øݾ]°Þ XbÝJóŠ¶ì. X|&Æ3— f(x) ≡ 0 (mod m) ݊`, —ÝÎ modulo m Ý!õ v, .h &Æ1 f(x) ≡ 0 (mod m) ݊ÝÍó`, —ÝÎ3 modulo m ìb9KÝ8

²!õvº”• f(x) ≡ 0 (mod m), ‚Ηb9KÍJóŠ.

9͍—¼:, &Æ©ŠŒ×Í modulo m Ý complete residue system S, Q¡Þ S Ý-ô××ñá f(x) , ::ø×°º¸ÿ f(x) ≡ 0 (mod m), £‚µ|0ÕXb ݊Ý. Ä9]°3 m œ
`µ•ÿ6@jÝ. .h&ÆTs"ק¡, ‹ 43

K§Š×°´©„Ý congruence equation ͊ݩP. Äѧø, &Æá¼×Í congruence equation3 modulo m ì͊ÝÍó‹9µÎ m.

Í@î, &ÆGµ#ÇÕ×°Š congruence equation Ý®ÞÝ. 3 modulo m ì 0 a ∈ Z ݶ°D-ôݮޯ@îµÎ3Š ax ≡ 1 (mod m) (Ç ax − 1 ≡ 0 (mod m)) 9×Í congruence equation. ã Proposition 3.2.5 á a õ m !²`, h congruence equation PŠ. ¨²î Proposition 3.2.3, &Æá¼ a õ m !²`h congruence equation 3 modulo m ìb°×Š.

A Lemma 3.4.2 ÎD¡ p βó` x<sup>2</sup> ≡ 1 (mod p)݊. h`ã Lemma 3.4.2 &

Æá p βó`bË͊, 5½Î x ≡ 1 (mod p) õ x ≡ −1 (mod p). &ÆèÄ m

βó`, 4Q x ≡ ±1 (mod m) ) x² ≡ 1 (mod m)9×Í congruence equation Ý Ë͊, ¬h congruence equation bb9yË͊. »A x² ≡ 1 (mod 15)݊µÎ x ≡ ±1 (mod 15) õ x ≡ ±4 (mod 15) 9 4 ͊. 9õ&Æ×!á×Í n g94P‹

9b n ͊!, T©½¥Œ.

×Í n gÝ@;ó94P‹9b n ͊Ýæ.Î. @;ó94P ôbXÛÝ t°æ§, 9Í槬à3Jó94P. Ä tPÎ×Ít{g4;ó 1 ÝJ;ó94P`, )àt°æ§. ãy&ƬmŠ×ÝP², 9…&ÆG"Dt PÎ×g94Pݵ.

Lemma 4.1.1. ƒ' f(x) Î×Í n g (n ≥ 1) ÝJ;ó94Pv a ∈ Z. JD3×Í n − 1 gÝJ;ó94P h(x) |C r ∈ Z ”•

f (x) = (x − a)h(x) + r.

Proof. E f(x) Ýgó n †ó.hû°. ƒ' f(x) Î 1 g94P, Ç f(x) = c1x + c₀,J

ƒ h(x) = c1 v r = ac1+ c₀,&Æÿ (x − a)h(x) + r = f(x).

Tàó.hû°, ƒ'Egó n < k ÝJ;ó94P g(x), /D3 n − 1 gÝJ;ó 94P h0(x) |C r0 ∈ Z ¸ÿ g(x) = (x − a)h0(x) + r₀. ¨Ê f(x) Ýgó n = k Ý, ôµÎ1 f(x) = ckx^k+ c_k−1x^k−1+ · · · + c₁x + c₀, Í ci ∈ Z v ck 6= 0. ƒ g(x) = f (x) − (x − a)c_kx^k−1,J g(x) = (ck−1+ c_ka)x^k−1+ · · · c₁x + c₀ Î×Ígóy k ÝJ;ó94P. Æàhûƒ'áD3×góy k − 1 ÝJ;ó94P h0(x) |C r₀ ∈ Z¸ÿ g(x) = (x − a)h0(x) + r₀. ôµÎ1 f(x) = (x − a)ckx^k−1+ (x − a)h₀(x) + r₀. ƃ h(x) = ckx^k−1+ h₀(x) |C r = r0,&Æb h(x) Î×Ígó k − 1 ÝJ;ó94

Pv r ∈ Z ”• f(x) = (x − a)h(x) + r. ¤

à Lemma 4.1.1, &Ɲ|Jÿ p Îײó`3 modulo p ì×Í n gÝ congruence equation t9b n ͊. Ä´&ÆmE×Í congruence equation ÝgóìÍL.

Definition 4.1.2. ƒ' f(x) = cnxⁿ+ · · · + c₁x + c₀ Î×ÍJ;ó94P, › m ∈ N.

(1) u m - cn,J&ÆÌ f(x) 3 modulo m ìÎ×Ígó (degree) n Ý94P.

4.1. Š Congruence Equation ÝæJ 45

(2) u m - cr ¬ m|ci, for r < i ≤ n, J&ÆÌ f(x) 3 modulo m ìÎ×Ígó r Ý94P.

AŒ×ÍJ;ó94P g(x) Í3 modulo m ìgó n, J&ÆÌ g(x) ≡ 0 (mod m)Î×Í n gÝ congruence equation.

ãhL&Æá¼u f(x) Î×Í3 modulo m ìgó n ÝJ;ó94P, b

 f(x) ͖ÝgóÎ
y n Ý. Ä&Ɲ|0Õ×Ígó n ÝJ;ó94P g(x) (»AÀœ f(x) | m JtÝ4) ¸ÿE×Jó a, /b f(a) ≡ g(a) (mod m).

X| f(x) ≡ 0 (mod m) ݊ºõ g(x) ≡ 0 (mod m) 8!. ãy&Æ©nT congruence equation݊, X|*¡ D¡×Í n gÝ congruence equation f(x) ≡ 0 (mod m) `, 

´P, &Ƶà#ƒ' f(x) Ýgó n.

Theorem 4.1.3 (Lagrange). ›ײó p |C×J;ó94P f(x). AŒ3 modulo p

ì f(x) ≡ 0 (mod p) Î×Ígó n Ý94P, J f(x) ≡ 0 (mod p) 3 modulo p  ì‹9b n ͊.

Proof. ´×P, &ƃ' f(x) = cnxⁿ+ · · · + c₁x + c₀,Í p - cn. &ÆE n †hû°.

´ f(x) = c1x + c₀ Î×gJ;ó94P`, ƒ' x ≡ a (mod p) Î f(x) ≡ 0 (mod p) Ý×͊. ¨¨ƒ' x ≡ b (mod p) ôÎ×͊, ùÇ c1a + c<sub>0</sub> ≡ c<sub>1</sub>b + c<sub>0</sub> (mod p). . gcd(p, c<sub>1</sub>) = 1, ã Lemma 3.2.4 ÿ a ≡ b (mod p). ôµÎ1 n = 1 `‹9b×͊.

àhûƒ' n < k `×Í n gÝ congruence equation ‹9b n ͊. ¨Ê n = k Ý. u x ≡ a (mod p) Î f(x) ≡ 0 (mod p) Ý×͊, ¿à Lemma 4.1.1 áD3

×Ígó k − 1 ÝJ;ó94P h(x) |C r ∈ Z ¸ÿ f(x) = (x − a)h(x) + r. µƒ

' x ≡ a (mod p) Î f(x) ≡ 0 (mod p) Ý×͊, Ç f(a) ≡ 0 (mod p), Þ a ‚áÿ f (a) = r ≡ 0 (mod p). ¨¨ƒ' x ≡ b (mod p) ôÎ×͊, Jã f(b) = (b − a)h(b) + r á (b − a)h(b) ≡ 0 (mod p). ðŽ, u b 6≡ a (mod p), Ç p - (b − a), Jã Lemma 1.4.2 á, p|h(b), ôµÎ1 x ≡ b (mod p) Î h(x) ≡ 0 (mod p) Ý×͊. .h&Æá¼ k g congruence equation f (x) ≡ 0 (mod p)݊ x ≡ a (mod p) T h(x) ≡ 0 (mod p) ݊.

Q‚ h(x) ≡ 0 (mod p) Î×Ígóy k Ý congruence equation, Ƶhû°ƒ'͋

9b k − 1 ͊, ÆÿJ f(x) ≡ 0 (mod p) ‹9b k ͊. ¤

t¡&Ægèø, ŠŠ congruence equation f(x) ≡ 0 (mod m) mފÝXbµ¶

ì¼, ׺ފ| x ≡ a (mod m) 9øÝP¶ì¼. Äb` Ý]-&ƺފ|

modulo½ÝóÝ]P¶ì. »AŠ x²≡ 1 (mod 8), &Æs¨XbÝóK”•, X|

Ý]-&Ɲ|ފ| x ≡ 1 (mod 2) ¶ì. Ċ¥Œ9ËP¶ì¡ &ÆèCŠÝ Íó`mèC3 modulo %‚ì݊ÝÍó. »A3h»&Ɲ|1 x²≡ 1 (mod 8) 3 modulo 8 ìb x ≡ 1, 3, 5, 7 (mod 8), 4 ͊, ô|13 modulo 2 ìb×͊.

4.2. ËÍðàÝ]°

&Æ+ÛËËðàÝ]°Þ×͛Ý congruence equation ;WŽ×FÝP, ¼O

Š.

39×;&ÆKƒ' f(x) = anxⁿ+ · · · + a₁x + a₀,Í ai ∈ Z,‚ m ∈ N Îכ

ÝÑJó. &Ɗ—¡ f(x) ≡ 0 (mod m) 9×Í congruence equation.

Ï×ËÎ9øÝ: AŒ d Î an, . . . , a₁, a₀ |C m ÝÑ2.ó. ôµÎ1&Ɲ|Þ a_i C m ¶W an= a⁰_nd, . . . , a₁ = a⁰₁d, a₀ = a⁰₀d|C m = m⁰d, Í9° a⁰_i∈ Z v m⁰ ∈ N.

ƒ g(x) = a⁰_nxⁿ+ · · · a⁰₁x + a⁰₀,&Ƽ"D f(x) ≡ 0 (mod m) C g(x) ≡ 0 (mod m⁰) 9Ë Í congruence equation  Ýn;.

Proposition 4.2.1. › m ∈ N C f(x) = anxⁿ+ · · · + a₁x + a₀, Í ai ∈ Z. ƒ' d Î an, . . . , a₁, a₀ C m ÝÑ2.óv an= a⁰_nd, . . . , a₁ = a⁰₁d, a₀ = a⁰₀d |C m = m⁰d. ƒ g(x) = a⁰_nxⁿ+ · · · + a⁰₁x + a⁰₀.

u x ≡ c (mod m⁰) Î g(x) ≡ 0 (mod m⁰) Ý×͊, JEŒ t ∈ Z, x ≡ c + m⁰t (mod m) f(x) ≡ 0 (mod m) ݊. ¨×]«, u g(x) ≡ 0 (mod m⁰) PŠ, J f(x) ≡ 0 (mod m) PŠ.

Proof. x ≡ c (mod m⁰) g(x) ≡ 0 (mod m⁰)Ý×͊, î m⁰|a⁰_ncⁿ+ · · · + a⁰₁c + a⁰₀. . hÿ m⁰d|a⁰_ndcⁿ+ · · · + a⁰₁dc + a⁰₀d,ôµÎ1 m|ancⁿ+ · · · a₁c + a₀. .h x ≡ c (mod m) Î f(x) ≡ 0 (mod m) Ý×͊.

EŒ t ∈ Z |C r ∈ N, ãy (c + m⁰t)^r = c^r+ rc^r−1m⁰t + · · · + rc(m⁰t)^r−1+ (m⁰t)^r,

&Ɲ|Þ (c + m⁰t)^r ¶W c^r+ m⁰λ_r, Í λr ∈ Z. .h

f (c + m⁰t) = a_n(c + m⁰t)ⁿ+ · · · + a₁(c + m⁰t) + a₀ = f (c) + a_nm⁰λ_n+ · · · + a₁m⁰λ₁. . d|ai,Æá dm⁰|a_im⁰,ôµÎ1 aim⁰ ≡ 0 (mod m). X|&Æÿ

f (c + m⁰t) ≡ f (c) ≡ 0 (mod m),

ôµÎ1EŒ t ∈ Z, x ≡ c + m⁰t ôºÎ f(x) ≡ 0 (mod m) Ý×͊.

¨×]«, u x ≡ c (mod m) f(x) ≡ 0 (mod m) Ý×͊, Ç m|ancⁿ+· · ·+a₁c+a₀, J m⁰|a⁰_ncⁿ+ · · · + a⁰₁c + a⁰₀. ôµÎ1 x ≡ c (mod m⁰) g(x) ≡ 0 (mod m⁰) Ý×͊.

.hu g(x) ≡ 0 (mod m⁰) PŠ, J f(x) ≡ 0 (mod m) ùPŠ. ¤ Proposition 4.2.1×å&Æ, AŒ x ≡ c (mod m⁰) Î g(x) ≡ 0 (mod m⁰) Ý×͊, J EŒ t ∈ Z, x ≡ c + m⁰t (mod m) -ºÎ f(x) ≡ 0 (mod m) Ý×͊. Ä9…ãy&

ƊÊ3 modulo m ݵ, œ9ŠÎ¥Ý. ¯@îu t ≡ t⁰ (mod d),Jã d|t − t⁰,

ÿ dm⁰|m⁰(t − t⁰). ôµÎ1 c + m⁰t ≡ c + m⁰t⁰ (mod m). .h&Æ©ŠÊ x ≡ c + m⁰t (mod m)Í 0 ≤ t ≤ d − 1, µ|Ý.

Proposition 4.2.1Þ×Í modulo m Ý congruence equation ;W×Í modulo f´

Ý m⁰ Ý congruence equation. 9ø×¼ãy3 modulo m⁰ 슁ÊÝó´K, TŒÞ

4.2. ËÍðàÝ]° 47

æ¼Ý®Þ;Ý. Q‚u an, . . . , a₁, a₀ õ m Î!²Ý, &Æ)Q|Ê modulo ´

ÝÂ::b^bŠ. ¯@î, &Æb|씌.

Lemma 4.2.2. › m ∈ N C×J;ó94P f(x). u m⁰|m v f(x) ≡ 0 (mod m⁰) P

Š, J f(x) ≡ 0 (mod m) ùPŠ.

Proof. ƒ' f(x) ≡ 0 (mod m) bŠv x ≡ c (mod m) Í׊, Ç m|f(c). ãy m⁰|m, á m⁰|f (c), ôµÎ1 x ≡ c (mod m⁰) f(x) ≡ 0 (mod m⁰) ׊. hƒ' f(x) ≡ 0 (mod m⁰)PŠë;, ÆÿJ f(x) ≡ 0 (mod m) PŠ. ¤ Lemma 4.2.2 õ Proposition 4.2.1 !3y Proposition 4.2.1 Þæ94P&;ó t|2.ó¡Ê modulo m⁰ Š, ‚v¿à͊ÿÕæ94P3 modulo m Š, ‚ Lemma 4.2.2¬^b;Ž94P, vGáæ94P3 modulo f´Ý m⁰ ìPŠ.ÿ æ94P3 modulo m ìPŠ. ¬P
¾\3 modulo m⁰ ìbŠÎ͝ÿ3 modulo m

ìbŠ, ‚vôP
.ÿŠP. Äu&Æ9Ê¿Í m Ý.óXÿÝ congruence equations, @@|Q&ÆÿáŠ. 9µÎ&Ɗ"DÝÏÞË]°.

9×ËðàÝ]°µÎÞ m ¶W².óÝ5Š, Ç m = pⁿ₁¹· · · pⁿ_r^r, Í9° pi 8²²ó. #½GŠ"DEXb i = 1, . . . , r, f(x) ≡ 0 (mod pⁿ_iⁱ)ŠÝµ, . &

Æb|씌.

Proposition 4.2.3. ƒ' m = pⁿ₁¹· · · pⁿ_r^r, Í9° pi 8²²óv f(x) ×J;ó9 4P. uD3 i ∈ {1, . . . , r}, ¸ÿ f(x) ≡ 0 (mod pⁿ_iⁱ) PŠ, J f(x) ≡ 0 (mod m) PŠ.

¨×]«, EŒ i ∈ {1, . . . , r}, x ≡ c (mod pⁿ_iⁱ) / f(x) ≡ 0 (mod pⁿ_iⁱ) ݊uv°u x ≡ c (mod m) f(x) ≡ 0 (mod m) ÍŠ.

Proof. ´, ãy pⁿ_iⁱ|m, .hà Lemma 4.2.2 á, u f(x) ≡ 0 (mod pⁿ_iⁱ) PŠ, J f (x) ≡ 0 (mod m)PŠ.

¨ƒ' x ≡ c (mod m) f(x) ≡ 0 (mod m) Ý×͊, ôµÎ1 m|f(c), ãyEŒ

i ∈ {1, . . . , r} /”• pⁿ_iⁱ|m, á pⁿ_iⁱ|f (c). .háEXbÝ i ∈ {1, . . . , r}, x ≡ c (mod pⁿ_iⁱ) f(x) ≡ 0 (mod pⁿ_iⁱ) ݊.

D, uEXb i ∈ {1, . . . , r}, x ≡ c (mod pⁿ_iⁱ) / f(x) ≡ 0 (mod pⁿ_iⁱ) ݊. Ç pⁿ_iⁱ|f (c). Jãy9° pⁿ_iⁱ ÎËË!²Ý, ¿à Proposition 1.2.7(2) á pⁿ₁¹· · · pⁿ_r^r|f (c),ùÇ

m|f (c). ÆÿJ x ≡ c (mod m) f(x) ≡ 0 (mod m) Ý×͊. ¤

Proposition 4.2.3×å&Æ, ub×Í pi ¸ÿ f(x) ≡ 0 (mod pⁿ_iⁱ)PŠ, £‚ f(x) ≡ 0 (mod m) µPŠ. ¬ÎAŒEXbÝ pi, f (x) ≡ 0 (mod pⁿ_iⁱ) /bŠ, ÎÍî f(x) ≡ 0 (mod m)bŠ÷? ŒnÎùÝ. 9Î. 4QEŒÝ pi Šÿ݊ÎÄ8!, ¬¿à|

¡º"DÝ»yõ§0Õ×Jó!`”• modulo p<sup>n</sup><sub>i</sub><sup>i</sup> ìN͊ÝP, .hã Proposition 4.2.3ÿá f(x) ≡ 0 (mod m) bŠ. nyhI |¡3"D»yõ§`&

ƺ1€.

4.3. ×gÝ Congruence Equations

&Æ"DtŽÝ×Ë congruence equation, ×gÝ congruence equation. &Æ޺ἠ͊ÝÍóCŠÝP.

› m ∈ N XÛ modulo m Ý×g congruence equation Ç ax ≡ b (mod m) 9ø

PÝ congruence equation, Í a, b ∈ Z v m - a. ´&Ƽ::A¢¾½×Í×gÝ congruence equationÎÍbŠ.

Proposition 4.3.1. › m ∈ N. Ê×gÝ congruence equation ax ≡ b (mod m), Í

 m - a. ƒ' d = gcd(m, a). J d|b uv°uh congruence equation bŠ.

Proof. . d = gcd(m, a) Æã Corollary 1.2.5 áD3 r, s ∈ Z ¸ÿ d = rm + sa.

¨ƒ' d|b, ÇD3 b⁰ ∈ Z¸ÿ b = b⁰d. .h b = b⁰d = b⁰rm + b⁰sa,Æuƒ x = sb⁰,J ax = asb⁰ = b − b⁰rm. ôµÎ1 m|ax − b, ÿJ x ≡ sb⁰ (mod m) ax ≡ b (mod m) 

׊.

D, u x ≡ c (mod m) ax ≡ b (mod m) ׊, Ç m|ac − b. ðŽ, D3 r ∈ Z

¸ÿ ac − b = mr, ôµÎ1 b = ac − mr. ¨ãy d = gcd(m, a), &Æb d|m v d|a, Æÿ

J d|b. ¤

ã Proposition 4.3.1 ÝJ€&Æá¼, › m ∈ N, v a, b ∈ Z. ƒ' gcd(m, a) = d v d|b. u r, s, b⁰ ∈ Z ”• d = rm + sa v b = b⁰d, J x ≡ sb⁰ (mod m) ax ≡ b (mod m) Ý×͊. Ä9¬îXb݊KµhÿÕ. ŠA¢0ÕXb݊÷? ¶ï|G&

ÆðàÝ]°µÎ"Dˊ Ýn;, ¿àáÝ×͊¼0ÕXb݊. #ì¼

&Ƽ: ax ≡ b (mod m) ͊ Ýn;.

Proposition 4.3.2. › m ∈ N, Ê×gÝ congruence equation ax ≡ b (mod m). ƒ ' d = gcd(m, a) vá x ≡ c (mod m) Î ax ≡ b (mod m) Ý×͊, JEŒ ax ≡ b (mod m) ݊ c⁰ Kº”• c⁰ ≡ c (mod m/d). D, EŒÝ t ∈ Z,

x = c +m dt ù ax ≡ b (mod m) Ý×͊.

Proof. ƒ' x ≡ c⁰ (mod m)ù ax ≡ b (mod m) Ý×͊, Jãyá x ≡ c (mod m) ׊, Æÿ ac ≡ b ≡ ac⁰ (mod m). .hã Proposition 3.2.3 á c ≡ c⁰ (mod m/d).

D, u c⁰ = c + (m/d)t, Í t ∈ Z, J ac⁰ = ac + (a/d)mt. . d = gcd(m, a), Æ á a/d ∈ Z, ôµÎ1 ac⁰ ≡ ac (mod m). Äá ac ≡ b (mod m), X|ÿJ ac⁰ ≡ b

(mod m). ¤

Proposition 4.3.2 ×å&ÆÊ congruence equation ax ≡ b (mod m). u x ≡ c (mod m)Î×͊, J͸݊/ c + (m/d)t 9øÝP, Í d = gcd(m, a) v t ∈ Z.

.há3 modulo m ì x ≡ c + (m/d), x ≡ c + 2(m/d), . . . , x ≡ c + (d − 1)(m/d) Kº

4.3. ×gÝ Congruence Equations 49

Î ax ≡ b (mod m) ݊. &ÆÞºJ€9°Š3 modulo m ì/8², ‚v3 modulo mìXb݊K 9°P, .há) Proposition 4.3.1 |C Proposition 4.3.2, &

Æb|씌.

Theorem 4.3.3. › m ∈ N, a, b ∈ Z Ê×gÝ congruence equation ax ≡ b (mod m).

ƒ d = gcd(m, a).

(1) u d - b, J ax ≡ b (mod m) PŠ.

(2) u d - b, J ax ≡ b (mod m), 3 modulo m ìb d ͊. vuá x ≡ c (mod m) ׊, J

x ≡ c +m

dt, t = 0, 1, . . . , d − 1 ax ≡ b (mod m) 3 modulo m ìXb݊.

©½2, a õ m !²`, EyXb b ∈ Z, ax ≡ b (mod m) /bŠ, v͊3 modulo m

ìΰ×Ý.

Proof. µ Proposition 4.3.1 |C Proposition 4.3.2, &Æ©yìŠJ€ ax ≡ b (mod m) u bŠ, J3 modulo m ìb d ͊. .h&ÆmJ€Ë¯: (×) 0 ≤ i, j ≤ d − 1 v i 6= j ` c + mi/d 6≡ c + mj/d (mod m) (Ah-ÿ 0 ≤ i ≤ d − 1 ` c + mi/d 3 modulo m ì/8²). (Þ) EŒ t ∈ Z, /D3 i ∈ {0, 1, . . . , d − 1} ¸ÿ c + mt/d ≡ c + mi/d (mod m) (Ah-ÿJXb݊@¶ c + mi/d, Í 0 ≤ i ≤ d − 1 ÝP).

ƒ' 0 ≤ i, j ≤ d − 1 v i 6= j. ´×P&ƃ' i > j, h` 1 ≤ i − j ≤ d − 1. u c + mi/d ≡ c + mj/d (mod m), Ç (m/d)i ≡ (m/d)j (mod m). ãy gcd(m/d, m) = m/d, Æã Proposition 3.2.3 á i ≡ j (mod m/(m/d)), Ç i ≡ j (mod d). ôµÎ1 d|i − j. h õ 1 ≤ i − j ≤ d − 1 ë;, ÆÿJ c + mi/d 6≡ c + mj/d (mod m).

¨á ax ≡ b (mod m) ݊/ c + mt/d, Í t ∈ Z 9øÝP. EŒ t ∈ Z, ã Theorem 1.2.1áD3 h, r ∈ Z ¸ÿ t = hd + r, Í 0 ≤ r ≤ d − 1. .hÿ

c + mt/d = c + m(hd + r)/d = c + mh + mr/d.

ƃ i = r, &Æb 0 ≤ i ≤ d − 1 v c + mt/d ≡ c + mi/d (mod m). ôµÎ1 ax ≡ b (mod m)݊/ c + mi/d, Í 0 ≤ i ≤ d − 1 9øÝP. ¤ ã Theorem 4.3.3 &Æáu ax ≡ b (mod m) bŠ, ©ŠŠŒÍ×͊, ̀݊µ

ÿÕ. ‹y0ŠÝ]°, tÝ Proposition 4.3.1 ÝJ€X+ÛÝ]°², ¯@î&Æ

|¿à Proposition 4.2.1 XèÝ]°¼Š. . h`u d = gcd(m, a), J d|b, ôµÎ 1 d Î a, b õ m Ý2.ó. ÆuÞ a, b, m 5½¶W a = a⁰d, b = b⁰dõ m = m⁰dÝ

P (Í a⁰, b⁰, m⁰ ∈ Zv gcd(m⁰, a⁰) = 1),¿à Proposition 4.2.1 &Æá|Š a⁰x ≡ b⁰ (mod m⁰) 9×Í congruence equation. ãy gcd(a⁰, m⁰) = 1,µ Proposition 3.2.5 áD3 e ∈ Z¸ÿ a⁰e ≡ 1 (mod m⁰). ÆÞ a⁰x ≡ b⁰ (mod m⁰) Ë\¶î e ÿ

x ≡ a⁰ex ≡ b⁰e (mod m⁰).

.hÿ x ≡ b⁰e (mod m⁰) a⁰x ≡ b⁰ (mod m⁰) Ý×͊, .‚ã Proposition 4.2.1 ÿ á x ≡ b⁰e (mod m) ax ≡ b (mod m) Ý×͊. ‹yh e (Ç a⁰ 3 modulo m⁰ ì ݶ°D-ô) u|0, ¿à Corollary 3.3.3 (Euler’s Theorem) 0Õ. &Æ:|ì Ý».

Example 4.3.4. &ƊŠ 16x ≡ 8 (mod 52). . gcd(52, 16) = 4 v 4|8, Æáh congruence equation ÄbŠ, v3 modulo 28 ìb 4 ͊.

´&ÆŠ 4x ≡ 2 (mod 13). ãy 4 × 10 ≡ 1 (mod 13), &Æÿá x ≡ 2 × 10 ≡ 7 (mod 13) 4x ≡ 2 (mod 13) Ý×͊. .‚ÿ x ≡ 7 (mod 52) 16x ≡ 8 (mod 52) Ý

×͊ (Ç 16 × 7 = 112 = 52 × 2 + 8).

‹ỳ݊, ãy 52/4 = 13 Ƶ Theorem 4.3.3 á3 modulo 52 ì x ≡ 7, 20, 33, 46 (mod 52) 16x ≡ 8 (mod 52) ÝXbŠ.

4.4. Chinese Remainder Theorem

ƒ' m = pⁿ₁¹· · · pⁿ_r^r Í pi 8²²óv f(x) Î×ÍJ;ó94P. Proposition 4.2.3

×å&ÆuEXb i ∈ {1, . . . , r}, f(x) ≡ 0 (mod pⁿ_iⁱ) /bŠvb!Š, J f(x) ≡ 0 (mod m)-bŠ. A¢0Õ!Š÷? »yõ§ (Chinese Remainder Theorem) ×å

&Æ©ŠÍ½2Þ f(x) ≡ 0 (mod pⁿ_iⁱ) ݊0Õ, µÿÕ!Š.

Theorem 4.4.1 (Chinese Remainder Theorem). ›×à m1, . . . , m_r ∈ N Í9° mi

/ËË!² (Ç i 6= j `, gcd(mi, m_j) = 1). JEŒÝ×à c1, . . . , c_r∈ Z /0Õ×

Jó c ¸ÿ

c ≡ c_i (mod m_i), ∀ i ∈ {1, . . . , r}.

Proof. Ý]-, &ƃ M = m1· · · m_r vEŒ i ∈ {1, . . . , r}, ƒ Mi = M/m_i.

Š¥Œ9… Mj õ mi b|ìÝn;: (1) u i 6= j, J mi|M_j. (2) gcd(M_i, m_i) = 1. 9… (1)ã Mj ÝL8*
œ|ÿá, ‹y (2) ´×P (Žð×ì mi Ý5), &ÆG mJ€ gcd(M1, m₁) = 1. ƒ' M1, m₁ !², ÇD3ײó p ¸ÿ p|M1 v p|m1. Q‚

µL M1 = m₂· · · m_r,Æã Corollary 1.4.3 áD3 i ∈ {2, . . . , r} ¸ÿ p|mi. ¬Î i 6= 1, µƒ' gcd(m1, m_i) = 1,Æ p|m1 v p|mi õ m1, m_i !²8ë;, ÆÿJ gcd(M1, m₁) = 1.

#ì¼&ƏŠ0Õ×à t1, . . . , t_r ∈ Z ¸ÿEXbÝ i ∈ {1, . . . , r}, t = c₁M₁t₁+ · · · + c_rM_rt_r

/”• t ≡ ci (mod m_i). Q‚E¢Ý×à t1, . . . , t_r ∈ Z |CכÝ i ∈ {1, . . . , r}, ã (1) (Ç mi|M_j for i 6= j) &Æ/b t ≡ ciM_it_i (mod m_i). Æ&ÆGm0Õ ti ∈ Z¸ÿ c_iM_it_i≡ c_i (mod m_i)ǝ. Q‚ã (2) (Ç gcd(Mi, m_i) = 1)|C Proposition 3.2.5 áD3 e_i ∈ Z ¸ÿ Mie_i≡ 1 (mod m_i),Æuƒ ti = e_i,Jÿ t ≡ ciM_ie_i ≡ c_i (mod m_i). .hEX b i ∈ {1, . . . , r}, &Æ0Õ ei ¸ÿ Mie_i ≡ 1 (mod m_i),ƒ c = c1M₁e₁+ · · · + c_rM_re_r, Jÿ c ≡ ci (mod m_i), ∀ i ∈ {1, . . . , r}. ¤

4.4. Chinese Remainder Theorem 51

Š¥Œ! 9° mi ÎËË!²`, ›ŒÝ c1, . . . , c_r Œÿ0Õ×ÍJó c

¸ÿ c ≡ ci (mod m_i) EXbÝ i ∈ {1, . . . , r} KWñ. »A m1 = 4, m₂ = 6`uÊ c<sub>1</sub> = 1, c<sub>2</sub> = 2, J0Õ×Jó c !`”• c ≡ 1 (mod 4) v c ≡ 2 (mod 6). 9Î.

u c ≡ 1 (mod 4) î c 4k + 1 ÝP, ÆÄ ó. Q‚u c ≡ 2 (mod 6), J c 6k + 2 P, Ä ‰ó. .h Q0Õ×JóÎóêΉó.

×¼1&Ɲ|Þ»yõ§:WΊA











x ≡ c₁ (mod m₁) x ≡ c₂ (mod m₂)

... ... x ≡ c_r (mod m_r)

9øÝÐñ]P. ×¼1Ðñ]PΊ0Õ×Í!Š!`Ð)9 r ÍPŽÆR

¼´p. ‚3 Theorem 4.4.1 ÝJ€,
|:Œ¢ó t1, . . . t_r Ý', µÎŠ.9 r ÍÐñÝP;W r Í}ñÝPͽŠŒ ti ¼, ŠQµŽŽÝ. &Ƽ::|ìÝ»

.

Example 4.4.2. › m1= 3, m₂ = 4, m₃ = 5|C c1 = 2, c₂= 1, c₃ = 3&ÆT0Õ

×Jó c ¸ÿ c ≡ ci (mod m_i), ∀ i ∈ {1, 2, 3}. ôµÎ10Õ c !`”•





c ≡ 2 (mod 3) c ≡ 1 (mod 4) c ≡ 3 (mod 5)

µï Theorem 4.4.1 ÝÐr°&Æb M1 = 20, M₂ = 15 |C M3 = 12. ´&Æ0Õ e₁ ∈ Z ¸ÿ M1e₁ ≡ 1 (mod m₁), Ç 20e1 ≡ 1 (mod 3), ôµÎ1”• 2e1 ≡ 1 (mod 3).

ãh0Õ e1= 2. !§&Ɗ0Õ e2, e₃ 5½”• 15e2≡ 1 (mod 4) (Ç 3e2 ≡ 1 (mod 4))

|C 12e3 ≡ 1 (mod 5) (Ç 2e3≡ 1 (mod 5)). ÿ e2 = 3õ e3 = 35½”•îP. ƃ

c = 2 × 20 × 2 + 1 × 15 × 3 + 3 × 12 × 3 = 233”• 233 ≡ 2 (mod 3), 233 ≡ 1 (mod 4) | C 233 ≡ 3 (mod 5).

G«èÄ, › m ∈ N, ƒ' m = pⁿ₁¹· · · pⁿ_r^r, Í pi 8²²ó. AŒ f(x) Î×Í J;ó94P, ŠŠ f(x) ≡ 0 (mod m), &Ɲ|ENÍ pi ÊŠ f(x) ≡ 0 (mod pⁿ_iⁱ).

AŒb×Í pi sß f(x) ≡ 0 (mod pⁿ_iⁱ)PŠÝµ, £‚µ Proposition 4.2.3 á f(x) ≡ 0 (mod m) PŠ. AŒN×Í pi /º¸ÿ f(x) ≡ 0 (mod pⁿ_iⁱ), Jµ Proposition 4.2.3 á,

mŠÐñ]P 









f (x) ≡ 0 (mod pⁿ₁¹) f (x) ≡ 0 (mod pⁿ₂²)

... ...

f (x) ≡ 0 (mod pⁿ_r^r)

b×!Šÿ f(x) ≡ 0 (mod m) ݊. ŠÐñ]ÎæpÝ, ‚»yõ§×å&

Ɲ|āʊÐñP, ͽފOŒ-ÿÕ!݊.

Corollary 4.4.3. ƒ' m = pⁿ₁¹· · · pⁿ_r^r, Í9° pi 8²²óv f(x) ×J;ó94 P. JEŒ i ∈ {1, . . . , r}, f(x) ≡ 0 (mod pⁿ_iⁱ) /bŠuv°u f(x) ≡ 0 (mod m) bŠ.

Proof. µ Proposition 4.2.3 á, AŒ f(x) ≡ 0 (mod m) bŠ, JEŒ i ∈ {1, . . . , r}, f (x) ≡ 0 (mod pⁿ_iⁱ)/bŠ.

¨ƒ'EŒ i ∈ {1, . . . , r}, f(x) ≡ 0 (mod pⁿ_iⁱ) /bŠv x ≡ ci (mod pⁿ_iⁱ) Í׊.

ãy9° pⁿ_iⁱ ÎËË!²ÝƵ Theorem 4.4.1 á, D3 c ∈ Z ”•EŒ i ∈ {1, . . . , r} / b c ≡ ci (mod pⁿ_iⁱ). ôµÎ1Œ i ∈ {1, . . . , r}, x ≡ c (mod pⁿ_iⁱ) f(x) ≡ 0 (mod pⁿ_iⁱ)

׊. Æ¿à Proposition 4.2.3 ÿá x ≡ c (mod m) f(x) ≡ 0 (mod m) ׊. ¤

&Ƶ¼:×Í9]«ÝŽ». 4Q9ìÝ»|à#‚óCÿՊŒ, ¬Î&

Æ©ÎT¿àh»¼ýŠ9…XàÝÃF, X|T
݊T½¥yA¢TàX.Ý ]°‚3yŠŒ ¢.

Example 4.4.4. &ƼŠ x² ≡ 1 (mod 15). µG«”Œá&Ɲ|5½Ê x² ≡ 1 (mod 3) C x² ≡ 1 (mod 5) ݊. . 3 õ 5 / ²ó, µ Lemma 3.4.2 á x ≡ ±1 (mod 3)õ x ≡ ±1 (mod 5) 5½ x²≡ 1 (mod 3) õ x² ≡ 1 (mod 5) Š. .h&Ɗ

0Õ|ìÝ°ÍÐñÝ congruence equation:

(1)

½ x ≡ 1 (mod 3) x ≡ 1 (mod 5) , (2)

½ x ≡ −1 (mod 3) x ≡ −1 (mod 5) , (3)

½ x ≡ −1 (mod 3) x ≡ 1 (mod 5) , (4)

½ x ≡ 1 (mod 3) x ≡ −1 (mod 5) .

(1) õ (2) &Ɯ|:Œ5½ãJó 1 õ −1 µ5½”• (1) õ (2). ‚ 11 |”

• (3), 4 |”• (4). X|ã Proposition 4.2.3 &Æá x ≡ 1, −1, 11, 4 (mod 15) K x² ≡ 1 (mod 15)݊. &Æ0Õ x²≡ 1 (mod 15)3 modulo 15 ìÝ 4 ͊, ¬î µ©b9 4 ͊. Ä
|Š•™J×ì3 modulo 15 ì@@Gb9 4 ͊.

3îÍ»&ƊŒ x²≡ 1 (mod 15)3 modulo 15 ìÝ 4 ͊¬ö@ÎÍ Gb9 4 ŠÎ. &Æá3¿à»yõ§`, ÎÍb̀݊. ôµÎ1 Theorem 4.4.1©×å&ƊÝD3P, ¬Î×å&ÆÎÍb̀Š. Q&ÆKἺbPM9Š,

¬Î̀݊A¢ÿá÷? &Æà×gðà݀]°, ::Ë͊ Ýn; ¢, Š QµÞXb݊¶ìÝ.

Theorem 4.4.5. ›×à m1, . . . , m_r∈ N Í9° mi /ËË!². ƒ M = m1· · · m_r, JEŒÝ×à c1, . . . , c_r∈ Z |ìÐñÝ congruence equation











x ≡ c₁ (mod m₁) x ≡ c₂ (mod m₂)

... ... x ≡ c_r (mod m_r)

3 modulo M ìD3°×Ý×͊. ¯@îu c ∈ Z ”•hÐñ congruence equation, JEŒ c⁰ ∈ Z ”• c⁰ ≡ c (mod M ) /º”•hÐñ congruence equation.

4.4. Chinese Remainder Theorem 53

Proof. Theorem 4.4.1JÿD3P, &ƊJ€3 modulo m1· · · m_r ì͊°×.

ƒ' c, c⁰∈ Z /”•|îÐñÝ congruence equation. ôµÎ1EŒ i ∈ {1, . . . , r}

&Æ/b c ≡ ci (mod m_i)v c⁰≡ c_i (mod m_i). .hEŒ i ∈ {1, . . . , r} /b mi|c − c⁰. Q‚9° mi ËË!², Æ¿à Proposition 1.2.11(2), &Æÿ m1· · · m_r|c − c⁰, Ç c ≡ c⁰ (mod M ).ôµÎ13 modulo M ì͊°×.

¨×]«, u c ”•Ðñ congruence equation v c⁰ ∈ Z ”• c⁰ ≡ c (mod M ), Jãy EŒ i ∈ {1, . . . , r}, mi|M ,Æá c⁰ ≡ c ≡ c_i (mod m_i). ùÇ c⁰ ”•hÐñ congruence

equation. ¤

»A3 Example 4.4.2 , &Æá¼ x = 233 ”•





x ≡ 2 (mod 3) x ≡ 1 (mod 4) x ≡ 3 (mod 5)

9×àÐñÝ congruence equation, X|ã Theorem 4.4.5 áŒÝJó c ”• c ≡ 233 ≡ 53 (mod 60)K|”•9×àÐñ congruence equation. QÝôGb”• c ≡ 53 (mod 60) ÝJ󺔕hÐñ congruence equation.

Theorem 4.4.5f Theorem 4.4.1 J. . 3 Theorem 4.4.1 &ÆGèCŠÝD3 P, ‚ Theorem 4.4.5 èCŠ3 modulo m1· · · m_r ìÎD3v°×Ý, ‚v.‚ã×

Š0ÕXb݊. bÝhºÞ˧5¼—, ‚à#—¡´JÝ Theorem 4.4.5 ¬ Ì Chinese remainder theorem. &ÆÞ˧5xŠÎ. úŸ»yõ§

ŠÝD3PCA¢0Õ׊.