Yaa doonaya inuu noqdo Milyaneer waxaa dhawaanahan jirtay su'aal yar oo wanaagsan oo Günther Jauch si cad u lahaa inuu ka fikiro: "Lambarku had iyo jeer waa la qaybin karaa \(4\) iyada oo aan wax ka dhimanayn haddii lambarka laga sameeyay labadiisa lambar ee ugu dambeeya ...?" - waana meesha saxda ah ee ay tahay inaad ka fikirto xisaab ahaan in yar intii aad naftaada u oggolaan lahayd in ay ku sasabtaan waxyaalihii quruxda badnaa. Sababtoo ah in kasta oo jawaabaha sida "waa xitaa", "ka kooban yahay \(0\) " ama " wadarta iskutallaabtu waxay la mid tahay \(4\) " dhawaq macquul ah jaleecada hore, jawaabta saxda ah waxay ku saleysan tahay hanti fudud oo ah nidaamkayaga jajab tobanle.
Tiro \(X\) waxaa loo qaybin karaa \(4\) haddii oo keliya haddii tirada ay sameeyeen labadeeda lambar ee ugu dambeeya loo qaybin karo \(4\) . Caddeyntu waxay si toos ah uga socotaa matalaadda jajab tobanle. Tiro kasta oo dabiici ah \(X\) waxaa si gaar ah loogu matali karaa qaabka
\[
X = 100 \cdot X' + X''
\]
qor meesha \(X''\) uu yahay lambarka laga sameeyay labada lambar ee ugu dambeeya, tusaale ahaan, \(0 \leq X'' < 100\) , iyo \(X'\) uu yahay qaybta hore ee lambarka. Tan iyo
\[
100 = 25 \cdot 4
\]
waxay khuseysaa, raacdaa
\[
X = 25 \cdot 4 \cdot X' + X''.
\]
Ku darista koowaad ee \(25 \cdot 4 \cdot X'\) had iyo jeer waxaa loo qaybin karaa \(4\) iyadoon loo eegin \(X'\) . Sidaa darteed, inta ka hartay \(X\) marka loo qaybiyo \(4\) kaliya \(X''\) ayaa khuseysa. Si rasmi ah ayaa loo muujiyey:
\[
X \equiv X'' \pmod{4}.
\]
Tani waxay khuseysaa gaar ahaan:
\[
4 \mid X \iff 4 \mid X''.
\]
Xeerarka kala qaybsanaanta ee la midka ah ayaa soo baxa marka awoodda \(10\) module ay noqoto mid aad u fudud. Si loo qaybiyo \(4\) qodobka muhiimka ah wuxuu ahaa in \(100 \equiv 0 \pmod 4\) Waxay xitaa noqotaa mid xiiso badan marka qiimaha \(1\) ama \(-1\) uu dhaco halkii uu ka ahaan lahaa \(0\) .
Tusaale caadi ah waa kala qaybsanaanta \(11\) .
\[
10 \equiv -1 \pmod{11}
\]
Haddii tani run tahay, qiimaha booska ee moduleka tirada jajab tobanlaha ah \(11\) had iyo jeer way isbedbeddelaan \(1\) iyo \(-1\) .
\[
X = a_0 + 10a_1 + 10^2a_2 + 10^3a_3 + \dots
\]
Sidaa darteed, waxay soo socotaa
\[
X \equiv a_0 - a_1 + a_2 - a_3 + \dots \pmod{11}.
\]
Tiradu waxay u qaybsan kartaa \(11\) haddii oo keliya haddii wadarta lambarradeeda isbedbeddelaya loo qaybin karo \(11\) Tusaale ahaan \(918082\) tani waa kiiska.
\[
2 - 8 + 0 - 8 + 1 - 9 = -22,
\]
maadaama \(-22\) loo qaybin karo \(11\) , \(918082\) sidoo kale loo qaybin karo \(11\) .
Xitaa ka sii qurux badan ayaa ah xeerka loogu talagalay \(7\) , \(11\) iyo \(13\) isku mar. Waxay qabtaa in
\[
1001 = 7 \cdot 11 \cdot 13
\]
iyo sidaas
\[
1000 \equiv -1 \pmod{7}, \qquad
1000 \equiv -1 \pmod{11}, \qquad
1000 \equiv -1 \pmod{13}.
\]
Haddii aad tiro u qaybiso baloogyo saddex ah laga bilaabo midig ilaa bidix, sidaas darteed waad ku dari kartaa oo kala jari kartaa baloogyadan si kala duwan.
\[
X = 123456789
\]
Haddaba, haddii la tixgeliyo,
\[
789 - 456 + 123 = 456.
\]
Lambarka asalka ah wuxuu leeyahay isla module-ka harsan ee \(7\) , \(11\) iyo \(13\) sida \(456\) Sidaa darteed, tiro aad u badan waxaa lagu beddeli karaa mid aad u yar iyada oo aan la beddelin qaybintiisa saddexdan lambar.
Kala qaybsanaanta \(37\) waxay sidoo kale leedahay qaab la yaab leh oo qurux badan.
\[
999 = 27 \cdot 37
\]
khuseysaa
\[
1000 \equiv 1 \pmod{37}.
\]
Marka loo qaybiyo \(37\) baloogyada saddexda ah si fudud ayaa loogu dari karaa. Tusaale ahaan, laga bilaabo
\[
99937
\]
wadarta guud
\[
99 + 937 = 1036.
\]
Halkaas
\[
1036 = 28 \cdot 37
\]
Haddii \(99937\) loo qaybin karo 37, markaas 99937 sidoo kale waxaa loo qaybin karaa \(37\) .
Xeerarkan oo kale waxay marka hore u muuqdaan inay yihiin xeelado tirooyin ah, laakiin ugu dambeyntii waa uun adeegsiga isla fikradda: beddelidda awoodaha waaweyn ee tobanka iyadoo la adeegsanayo hadhaaga fudud ee modulo tirada su'aasha laga qabo. Tani waxay u beddeshaa tiro jajab tobanle ah oo weyn xisaabin fudud oo ku lug leh iswaafajin. Taasi waa sababta saxda ah ee xeerarka kala-qaybinta noocaas ahi ay uga badan yihiin khiyaamo xisaabeed oo keliya; waxay u taagan yihiin dhimista hadhaaga modulo. \(10^k\): Qaabka su'aalaha, waxay u muuqdaan sidii dabinno garasho oo yaryar, laakiin waxay si toos ah u horseedaan fikrad la yaab leh oo qurux badan oo ku jirta aragtida tirada.