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 a

 

n

an = a · a · a · ... · a   de n−ori

 

an · am =  an+m

an : am = an−m (n > m)

(an)m = anm

a0

= 1 (a = 0)

an · bn = (a · b)n

an : bn = (a :  b)n

 

 

n > m ⇔  an > am

 

a > b ⇔  an > bn

 

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   2105

 375

   

 105 = 7 · 15

   75 = 5 · 15

 

2105 = 27·15 = (27)15 = 12815

 

375 = 35·15 = (35)15 = 24315

 128  <  243

   12815 < 24315

 2105 < 375

   

391  

945

   

9 = 32

 

945 =

3245

= 390

 

91  >  90

 

391 > 390

 

391 > 945

   355

 536

   

 

 

355 > 354 (1)

 

354 > 536

 

354 = 33·18 = (33)18 = 2718

 

536 = 52·18 = (52)18 = 2518

 

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2718 > 2518

 

354 > 536 (2)

 

355 > 536

 

 

 

 

 

 142011 + 1

 

 142011 + 1

 

 

22011 + 5  

 22011 + 5

 

 u(n)

   n

   n p

 n

 

 p  

  n

   p

u(574 p) = u(4 p)

  u(a · b) = u (u(a) · u(b))    u(a + b) =  u (u(a) + u(b))

u(274 · 379) = u(4 · 9) = 6

   u(274 + 379) =  u(4 + 9) = 3

 u(1n);   u(2n);   u(3n)

 

u(4n);   u(5n);   u(6n);   u(7n);   u(8n);   u(9n);   u(0n)    n  

 u(1n)

   u(0n)

 

u(1n) = 1    u(0n) = 0

 u(5n)

   u(6n)

 u(51) = 5;   u(52) = 5;   u(53) = 5

 

u(5n) = 5

 u(6n)

   u(61) = 6;   u(62) = 6;   u(63) = 6

 

u(6n) = 6

 

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 2n

 u(21) = 2;   u(22) = 4;   u(23) = 8;   u(24) = 6

 u(24) = 6

   u(6n) = 6

 n

   n = 4 · k + r

   r

 u(2n) =  u(24·k+r) =  u(24·k · 2r) =  u

(24)k · 2r

 =  u(16k · 2r) =

u(6k · 2r) = u(6 · 2r)

 

u(2n) = u(24k+r) = u(6 · 2r)

  r

  u(22011) = u(24·502+3) = u(6 · 23) = 8

 u(3n);   u(7n)

   u(8n)

 

u(34) = 1;   u(74) = 1    u(84) = 6

 

u(3n) = u(34k+r) = u(1 · 3r) = u(3r)

u(7n) = u(74k+r) = u(1 · 7r) = u(7r)

u(8n) = u(84k+r) = u(6 · 8r)

  r

 4n

  u(41) = 4    u(42) = 6

   u(42) = 6

 n    n = 2 · k + r    r

 u(4n) =  u(42·k+r) =  u(42·k · 2r) =  u

(42)k · 4r

 =  u(16k · 4r =

u(6 · 4r)

 

u(4n) = u(42k+r) = u(6 · 4r)

 r

 9n

 u(92) = 1

u(9n) = u(92k+r) = u(9r)

 r

   a = 22010 + 32011 + 42012

 

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u(22010) = u(24·502+2) = u(6 · 22) = 4

u(32011) = u(34·502+3) = u(33) = 7

u(42012) = u(42·1006) = u(6 · 40) = 6

 

u(a) = u(4 + 7 + 6) = 7

   107103 +103107

 

 

u(107103) =  u(7103) = u(74·25+3) = u(73) = 3

 

u(103107) =  u(3107) = u(34·26+3) = u(33) = 7

 

u(107103 + 103107) = u(3 + 7) = 0

 107103 + 103107

 

   n

 

A = 22n+4 · 9n + 22n · 9n+1

   

 

A   = 22n · 24 · 9n + 22n · 9n · 9 = 22n · 9n ·

24 + 9

  = 22n · 9n · 25 =

(2n)2 ·

32n

· 52 = (2n · 3n · 5)2

  A

   

a = 63n + 7n+1 · 32n+1 − 21n · 3n+2

   

a =

32 · 7n

+7n·7·32n·3−(3·7)n·3n·32 = 32n·7n+7n·32n·21−32n·7n·9 =32n · 7n · (1 + 21 − 9) = 32n · 7n · 13

 

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 a

    a  = 3454 − 3453 − 3452  

b = 5340 

 

    a  

a = 3452 ·

32 − 3 − 1

 = 3452 · 5

 3452 · 5

   5340

 3452

 5339

 

3452 = 34·113 =

34113

= 81113

 

5339 = 53·113 =

53113

= 125113

 

81 <  125

 

81113 < 125113

 b

   a

  414; 512; 711; 117

 

   

414 =

2214

= 228 =

274

= 1284 (1)

 

512 =

534

= 1254 (2)

 

414 > 512 (∗)

 

1254 > 1214

1214 =

1124

= 118

 

118 > 117

 

512 > 117 (∗∗)

 

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711 = 79 · 72 = 733 · 72 = 3433 · 49

 

3433 · 49  >  2563 · 16

 

2563 · 16 =

283· 24 = 224 · 24 = 228 = 414

 

711 > 414 (∗ ∗ ∗)

117; 512; 414; 711