六年级奥数课堂：数论的方法技巧之一

六年级奥数课堂：数论的方法技巧之一

1abb0qr

a=bq+r0≤rb

qr

r=0a=bqabb|abaab

2a|cb|cabab|c

31n

p1p2pka1a2ak1n

4n1

dn=a1+1a2+1ak+1

5nn+1xyx≤y-1

1n=an10n+an-110n-1+…+a0

2a=bq+r

42n=2mtt

1 11411019983

a3a2a1a0

1000a3+100a2+10a1+a0

10

10a3+a2+a1+a0=10a3+10a2+10a1+10a0

10

990a3+90a2-9a0=1998

110a3+10a2-a0=222

a0=8a2=1a3=2

218

a+b+c14

3 123100018

abcd4

a+b+c=18ma+b+d=18n

mn

c-d=18m-n

21818r

a=18a1+rb=18b1+rc=18c1+r

a1b1c1

a+b+c=18a1+b1+c1+3r

18|a+b+c18|3r6|rr=06121000=55×18+101210006244299656318

4 N5491N10

N

N572=49a3≥1a4≥2ak

a1+1a2+1an+1=10

a3+1≥2a4+1≥310=2×5

a1+1=a2+1=a5+1=…=an+1=1

a1=a2=a5=…an=0N257a4+1≥32

a3+1a4+1=10

a3+1=5a4+1=2a3+1=2a4+1=5N=52-1×75-1=5×74=12005

5 N123199819992000N21

210=1024211=2048200020002101024=210N102

5

6 11

900

xyz1110

x2+y2+z2≤10

1≤x≤30≤y≤30≤z≤3

100101102103110111112

120121122130200201202

211212220221300301310

100101

7 N235352NN2000

N

N|100N=10202550

N|1000N=100125200250500

4NN=1000125020002500500010001250

14

1k10k

2

8 3103331315233

13+15+23=5151=3×17

171317 3 31715

1610 179 188

2510 269 278

3410 359 368

377 (11)449 (12)458

(13)467 (14)557 (15)566

3+5+5=133+3+9=155+9+9=23

3359

9 12

100790919293949596

111312712

114115116117118119120

121122123124125126

12122334121312

m+2m+3m+1312m231312

2m为2341312m+2m+3m+4m+132341312

:

13+213+313+13

n=1×2×3×…×n12

m+1+2m+1+3m+1+m+1m

10 1005

1

2

3

4

5

1999

1711131131737

211133173

3111731

41173

51731

67311

73117

81173

41999=4×499+3199973117

11 100

N

1N=2aa=01232a

2N=2a+mm2a2m

N=100100=26+362×36=7272

12313111

12 12340

1111

223

1

2

3133-1=2

33

44133+1

55

9-3+1=5

9131+3+9=13

14

14+13=27

1+3+9+27=40

133233

13