x p v p (x) x p p-adic valuation of x v 2 (8) = 3, v 3 (12) = 1, v 5 (10000) = 4, x 8 = 2 3, 12 = 2 2 3, = 10 4 = n a, b a

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1 . x p v p (x) x p p-adic valuation of x v (8) =, v () =, v 5 () =, x 8 =, =, = = 5. n a, b a b n a b n a b (mod n) (mod ), 5 (mod ), (mod 7),

2 a b = 8 =, 5 = 8 = ( ), = = 7 ( ),. Z n a b (mod n) a n b n (mod 7) 7 7 n n n n Z n = {,,,, n, n } Z n Z n + n a + n b = a + b n Z n n a n b = a b n + 5 = ( + = 7 5 ) = 5 ( = 5) 7 = ( = 7 ) = ( = )

3 n Z n + n n n +. Z n Z 5 5 =, 5 5 = 5 =, = 5 = x y =, =, =, = a a a a a a. Z 5 a a a a a a 5 a a 5 5 5

4 . Z 7 a Z 5 = 5 Z 7 = 7. (Fermat ) p n Z p (p ) Z p n p = ] Golomb Suppose we have beads in n different colors, and we wish to make necklaces using exactly p beads. First we put p beads on a string. Since each of the beads can be chosen in n ways, there are n p possible strings. For each of the n colors, there is one string entirely of that color. We throw these away, leaving n p n strings. We will join the two ends of each of these strings to form necklaces. But we observe that if two strings differ only by a cyclic permutation of the beads, the resulting necklaces will be indistinguishable. Since there are p cyclic permutations of the p beads on a string, the number of distinguishable necklaces is (n p n)/p, which must therefore be an integer. p = 5 n = 5 A B 5 = AAAAA AAAAB AAABA AABAA ABAAA BAAAA AAABB AABBA ABBAA BBAAA BAAAB AABAB ABABA BABAA ABAAB BAABA AABBB ABBBA BBBAA BBAAB BAABB ABABB BABBA ABBAB BBABA BABAB ABBBB BBBBA BBBAB BBABB BABBB BBBBB

5 A B p Z p a a k = k ord p (a).. Z 5 Z 7 Z 7,,, a. p Z p p ] a Z p p a k k p p k p = qk + r r < r < k a a p = a qk+r (.) Fermat a qk+r = a qk a r = (a k ) q a r = q a r = a r a k a k = (.) a r = a k k r a r = 5

6 k p. a Z p k n a n = k n ] p n. p f : Z p Z p f(x) = x G p = (V p, E p ) V p = Z p, E p = {(a, f(a)); a Z p} p p G

7 G 5 G 7 5 7

8 G G a Z p {a n} a = a, a = f(a ), a = f(a ),, a k+ = f(a k ) (k ) 8

9 a f a.a. p = a = Z a =, a = f(a ) = f() = =, a = f(a ) = f() = =, a = f(a ) = f() = = 9, a = f(a ) = f(9) = 9 =, a 5 = f(a ) = f() = = 9 a a a = a = a = tail number t(a) cycle number c(a) a = Z t() =, c() = 7 Z t(7) =, c(7) =. tale number cycle number. a Z p ord p (a) = e l e, l l t(a) = e, c(a) = ord l () 9

10 ] a t(a) = t, c(a) = c f t (a) = f t+c (a) (.) t, c (.) f a t = a t+c (.) a t a t+c a t = a t+c t = a t ( c ) = (.). a ord p (a) t ( c ) ord p (a) = e l e l t ( c ) ell e t, (.) l ( c ) (.5) (.) e t e = t (.) e < t (.7) (.5) e l e ( c ) a e ( c ) = (.) a e+c a e = a e+c = a e f e+c (a) = f e (a) tail number t t f k+c (a) = f k (a) k (.7) t (.) (.5) c (mod l) c ord l () c = ord l () ord l () = d c d c = d c > d modl d d (mod l) d l e l e ( d ) a e ( d ) =

11 f e+d (a) = f e (a) cycle number c f e+m (a) = f e (a) m c > d c = d p Z p p r Z p Z p. F : Z p Z p F (a) = r a F ] a, b Z p F (a + b) = F (a) F (b) (.) F (a + b) = r a+b = r a r b = F (a) F (b) (.) F F (a) = F (b) a = b F (a) = F (b) F r a = r b (.) a b (.) (r ) a r a (r ) a = (r r ) a = a =

12 b a {}}{{}}{ r b (r ) a = ( r r) ( r r ) = r b a r b a = a, b Z p b a p r p b a = a = b F F p F. Z p q : Z p Z p Z p m : Z p Z p F m = q F (.) ] a Z p (.) (F m)(a) = F (m(a)) = F (a) = r a = (r a ) = q(r a ) = q(f (a)) = (q F )(a) (.) Z p F Z p

13 Z m. Z k Z m m Z Z

14 Z Z

15 Z binary tree. Z k x = a a Z k () a () a a = b x = b, b + k ] () Z k x = a x a (mod k ) x a k a x a k () a = x = x (mod k ) x k x = x = k x = a (.) x x = a (.5) (.) (.5) (x x ) = x x =, k 5

16 x = x, x + k a = b x = b Z k. Z m. m d Z m d φ(d) m = d m φ(d) φ(d) d d.5 m d m = dd {x Z m ; dx = } = {x Z m ; d x} (.) ] x d x x = d x Z m dx = d(d x ) = (dd )x = mx = x x Z m dx = dx m dx = my y Z dx = dd y d x = d y x

17 . x Z m x m x m ] (.) G d M d x d m = dd x G d (.) x M d x d d < m d = m/d > x m d m = dd x M d (.) x G d x d m..7 x Z m x m x m.8 Z m m φ(m).5 G d = M d M d.9 f : Z d M d ( Z m )) f(a) = d a f 7

18 ] a, b Z d f(a+b) = d (a+b) = d a+d b = f(a)+f(b) f M d d a a < d f M d m/d = d Z d d f f. ] Z m d O d Z d m Z m = d m O d (.7) O d G d.5 G d = M d O d M d.9 M d Z d O d Z d d.8 φ(d) O d φ(d) (.7) m = d m φ(d). Z m m Z m D m Z m D G add. m Z m D ] m gcd(, m) = x + my = x, y mod m x = Z m x Z m E : Z m Z m E(a) = x a D E = E D = id D 8

19 . m Z m G add tail a Z m t(a) = ] a tail number t cycle number c D t (a) = D t+c (a) t > D(D t (a)) = D(D t+c (a)) (.8) a = D t (a), a = D t+c (a) a a cycle a a cycle a a (.8) D(a ) = D(a ) D.. (.7) Z m = d m O d O d D. m m d Z m d O d D D(O d ) O d ] a O d a O d d(a) = (da) = a d d d (a) = (d )a = a d d d d m d d d = d a O d 9

20 Z m D O d. m m d ord d () = l O d cycle l ] d G d.5 G d = M d.9 M d Z d Z d d.7 Z d = {a Z d; gcd(a, d) = } a Z d orb(a). t(a) = a cycle cycle number c D c (a) = a (.9) cycle number D c (a) a ( < c < c) (.) (.9) c a a (mod d) ( c )a (mod d) ( c )a d a d c d (.9) c (mod d) (.) (.) c (mod d) ( < c < c) (.) ord d () = c

21 Z Z 5 Z 7 5 Z 9

22 7 8 Z Z Z 5

23 Z Z 9

24 Z Z

25 Z Z 7 5

26 Z m, m a, a { x a (mod m ) x a (mod m ) mod m m

27 ] m m m u + m u = u, u mod m m u (mod m ) (.) mod m m u (mod m ) (.) x = m u a + m u a (.5) x = m u a + m u a m u a (mod m ) a (mod m ) ( (.)) x = m u a + m u a m u a (mod m ) a (mod m ) ( (.)) (.5) x ch : Z mm Z m Z m ch(x) = (r m (x), r m (x)) r m (x) x m ch ch m m ch x 7

28 .5 m, m ch : Z mm Z m Z m ch(x) = (r m (x), r m (x)) r m (x) x m ch. (.5).5 ch.5 Z m m Z m m Z m = Z Z m Z n n D n Z Z m (t, a) D m (t, a) = (D (t), D m (a)) (.). (t, a) Z Z m : D m ({(t, a)}) = { {(, E m (a)), (, E m (a))} t ϕ t = E m. D m : Z m Z m ] Z (.) D (t) t = Dm ({(t, a)}) = 8

29 ϕ D m ({(, a)}) = {(s, b) Z m; D (s) =, D m (b) = a} = {(s, b) Z m ; s =, b = a} = {(, E m (a)), (, E m (a))} Z m.7 m G add m Gadd m tail ] V = {(, a); a Z m }, E = {((, a), (, D m (a)); a Z m } G = (V, E ) G add m Vm add Gadd m \ V = {(, a); a Z m } D m (, a) = (, a) V m m G add m Gadd m 9

30 G add G add (,) (,) (,) (,) (,) (,)

31 G add 5 G add (,) (,) (,),) (,) (,) (,) (,) (,) (,)

32 G add 7 5 G add (,) (,) (,) () (,) (,),) (,) (,) (,) (,) (5),) (,5)

33 G add G add 8 (,5) (,7) (,),5) (,),) (,7) (,8) (,) (,8) (,) (,) (,) (,) (,) (,) (,) (,)

34 G add G add (,9) (,) (,7) (,) (,7) (,9) (,) (,5) (,) (,) (,5) (,8) (,) (,) (,) (,) (,8) (,) (,) (,) (,) (,). Z m m Z m m

35 Z m = Z Z m Z Z m (t, a) D m (t, a) = (D (t), D m (a)) (.7)..8 Z k D k a Z k D k ({a}) = { ϕ {b, b + k } a a a = b.9 (t, a) Z Z m : Dm ({(t, a)}) = {(, E m (a)), (, E m (a))} t = {(, E m (a)), (, E m (a))} t = ϕ t =, E m. D m : Z m Z m ].8.. f : X Y f : X Y F : X X Y Y F (x, x ) = (f(x ), f(x )) ((x, x ) X X ) 5

36 B Y, B Y F (B B ) = f (B ) f (B ) (.8). ] (x, x ) F (B B ) F (x, x ) B B (f(x ), f(x )) B B f(x ) B f(x ) B x f (B ) x f (B ) (x, x ) f (B ) f (B ) (.8) (.8). (t, a) Z Z m D m ({(t, a)}) = D ({t}) D m ({a}) (.9) D m. E m D m ({a}) = {E m (a)} D ({t}).8.9 Z m

37 . G v G add k v attaching V E V = V (G) (V (G add k ) \ {}) E = E(G) (E(G add k ) \ {(k, ), (, )}) {( k, v)} G add k v G G G add G add k G G add k G k. m G add m Gadd G add m ] V = {(, a); a Z m }, E = {((, a), (, D m (a)); a Z m } G = (V, E ) G add m Gadd m V (, a) (a Z m ) V add m \ V {(, a); a Z m } D m (, a) = (, a) V m m G add m Gadd m 7

38 G add G add (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) 8

39 G add 5 G add (,) (,) (,) (,) () (,) (,),) ( (,) () (,) (,) (,) (,) (,),) (,) (,) (,) 9

40 G add 7 5 G add 8 (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,5) (,) (,5) (,) (,) (,) (,) (,5) (,) (,5) (,) (,)

41 G add G add (,7) (,7) (,5) (,8) (,) (,) (,8) (,) (,7) (,) (,) (,5) (,5) (,) (,5) (,7) (,8) (,8) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,)

42 G add G add (,) (,) (,) (,) (,) (,5) (,) (,7) (,8) (9) (,) (,) (,) (,) (,) (,) (,5) (,) (,7) (,8) (,9) () (,) (,) (,) (,) (,) (5) (,) (,7),8) (,9) (,) (,) (,) (,) (,) (,) (,5) (,) (,7) (,8) (,9) (,)

43 .7 Z k m k m n G add n n. n v (n) = k n k n = k m m n = m k = Z k m = Z k Z m Z k Z m (t, a) D k m(t, a) = (D k(t), D m (a)) (.).8 { D ϕ a ({a}) = k {b, b + k } a a = b.9. (t, a) Z k Z m : D ({(t, a)}) = k m { ϕ v (t) = {( t, E m(a)), ( t + k, E m (a))} v (t) E m D m : Z m Z m

44 8 7 7: Z 7 () Z 7 Z 7 () Z Z Z 9 Z Z 9 Z. (.5) () Z G add Z 5 Z Z 9 () Z () () () Z 7 Gmult 7 Z 7

AtCoder Regular Contest 073 Editorial Kohei Morita(yosupo) A: Shiritori if python3 a, b, c = input().split() if a[len(a)-1] == b[0] and b[len(

AtCoder Regular Contest 073 Editorial Kohei Morita(yosupo) 29 4 29 A: Shiritori if python3 a, b, c = input().split() if a[len(a)-1] == b[0] and b[len(b)-1] == c[0]: print( YES ) else: print( NO ) 1 B: