AMC中的数论问题
1：Remembertheprimebetwee
1to100：
23571113171923297379838991
31374143475359616771
2：Perfect
umber
Let
P
istheprime
umberif
2p1
isalsotheprime
umberthe

2p
12
p1
is
the
perfect
umberForexample628496
3Let
abca0isthreedigitali
tegerif
a3b3c3
The
the
umber
iscalledDaffodils
umberThereareo
lyfour
umbers
153370371407
Let
abcda0isfourdigitali
tegerif
a4b4c4d4
The
the
umber
iscalledRoses
umberThereareo
lythree
umbers
163482089474
4：TheFu
dame
talTheoremofArithmetic
Every
atural
umber
ca
bewritte
asaproductofprimesu
iquelyuptoorder
k
∏piri
i1
5：Supposethataa
dbarei
tegerswithb0The
thereexistsu
iquei
tegersqa
drsuchthat
0≤rba
dabqr
6：1GreatestCommo
DivisorLetgcdabmaxd∈Zdaa
ddb
Fora
yi
tegersaa
dbwehavegcdabgcdbagcd±a±bgcdabagcdabaForexamplegcd15060gcd6030gcd30030
2Leastcommo
multipleLetlcmabmi
d∈Zada
dbd3Wehavethatabgcdablcmab
f7：Co
grue
cemodulo
Ifabmqm0the
wecallaco
grue
cebmoduloma
dwerewriteabmodm
1Assumeabcdmk∈Zk0m≠0Ifa≡bmodmc≡dmodmthe
wehave
acbdmodmacbdmodmakbkmodm
2Theequatio
ax≡bmodmhasasolutio
ifa
do
lyifgcdamdividesb
8：Howtofi
dtheu
itdigitofsomespeciali
tegers
1Howma
yzeroatthee
dof
Forexamplewhe
100LetNbethe
umberzeroatthee
dof100the

N

1005

10025

100125

20
4

24
2a
ZFi
dtheu
itdigita
Forexamplewhe
100a3
9：Pali
dromesuchas83438isa
umberthatremai
sthesamewhe
itsdigitsarereversed
Therearesome
umber
oto
lypali
dromebut112121，222484，11414641
1Somespecialpali
drome
that
2isalsopali
dromeForexample
121112121111212321111121234321
111111111212345678987654321
2Howtocreateapali
dromeAlmosti
tegerplusthe
umberofitsreverseddigitsa
drepeatitagai
a
dagai
The
wegetapali
dromeForexample
87781651655617267266271353135335314884
Butwhethera
yi
tegerhasthisPropertyhasyettoprove3Thepali
dromeequatio
mea
sthatequatio
fromlefttorighta
drighttoleftitallsetup
fForexample：
1242242112231132211121241388888831421211
Letaba
dcdearetwodigitala
dthreedigitali
tegersIfthedigitssatisfytheacbedced9the
abcdeedcba
10Featuresofa
i
tegerdivisiblebysomeprime
umberIf
iseve
，the
2

一个整数
的所有位数上的数字之和是3（或者9）的倍数，则
被3（或者9）整除
一个整数
的尾数是零，r