Ma ka ʻāpana hou o "Who Wants to Be a Millionaire ?", aia kahi nīnau liʻiliʻi maikaʻi a Günther Jauch i noʻonoʻo pono ai: "Hiki ke māhele ʻia kahi helu e \(4\) me ke koena ʻole inā ʻo ka helu i hoʻokumu ʻia mai kona mau huahelu ʻelua hope loa he...?" - a ʻo kēia kahi e pono ai ʻoe e noʻonoʻo makemakika no kekahi manawa, ma kahi o ka hoʻowalewale ʻia e nā mea hoʻowalewale nani. No ka mea, ʻoiai nā pane e like me "he like", "aia ka \(0\) , a i ʻole "ʻo ka huina o nā huahelu he \(4\) " ke kani kūpono i ka nānā mua ʻana, aia ka pane pololei i kahi waiwai maʻalahi o kā mākou ʻōnaehana decimal.
Hiki ke māhele ʻia kahi helu \(X\) e \(4\) inā a inā wale nō inā hiki ke māhele ʻia ka helu i hoʻokumu ʻia e kona mau huahelu hope ʻelua e \(4\) . Ma hope pololei mai ka hōʻike decimal. Hiki ke hōʻike kū hoʻokahi ʻia kēlā me kēia helu kūlohelohe \(X\) ma ke ʻano.
\[
X = 100 \cdot X' + X''
\]
e kākau i kahi \(X''\) ka helu i hoʻokumu ʻia mai nā huahelu ʻelua hope loa, ʻo ia hoʻi, \(0 \leq X'' < 100\) , a \(X'\) ka ʻāpana ma mua o ka helu. No ka mea
\[
100 = 25 \cdot 4
\]
pili, hahai
\[
X = 25 \cdot 4 \cdot X' + X''.
\]
ʻO ka addend mua \(25 \cdot 4 \cdot X'\) e hiki ke māhele mau ʻia e \(4\) me ka nānā ʻole i \(X'\) . No laila, no ke koena o \(X\) i ka wā i māhele ʻia e \(4\) ʻo \(X''\) wale nō ka mea pili. Hōʻike maʻamau ʻia:
\[
X \equiv X'' \pmod{4}.
\]
Pili kēia ma ke ʻano kūikawā:
\[
4 \mid X \iff 4 \mid X''.
\]
Kū mai nā lula mahele like i ka wā e lilo ai kahi helu i mana o \(10\) modulo i mea maʻalahi loa. No ka mahele ʻana e \(4\) ʻo ke kumu koʻikoʻi ʻo \(100 \equiv 0 \pmod 4\) Lilo ia i mea hoihoi loa ke kū mai nā waiwai \(1\) a i ʻole \(-1\) ma kahi o \(0\) .
ʻO kahi laʻana maʻamau ka puʻunaue ʻana e \(11\) .
\[
10 \equiv -1 \pmod{11}
\]
Inā he ʻoiaʻiʻo kēia, ʻo nā waiwai wahi o kahi helu decimal modulo \(11\) e hoʻololi mau ana ma waena o \(1\) a me \(-1\) .
\[
X = a_0 + 10a_1 + 10^2a_2 + 10^3a_3 + \dots
\]
No laila, ua hahai mai
\[
X \equiv a_0 - a_1 + a_2 - a_3 + \dots \pmod{11}.
\]
Hiki ke puʻunaue ʻia kahi helu e ka \(11\) inā a inā wale nō e hiki ke puʻunaue ʻia ka huina o kona mau huahelu ʻē aʻe e ka \(11\) No \(918082\) no ka laʻana, penei ke ʻano.
\[
2 - 8 + 0 - 8 + 1 - 9 = -22,
\]
a no ka mea hiki ke māhele ʻia ʻo \(-22\) e \(11\) , hiki ke māhele ʻia hoʻi ʻo \(918082\) e \(11\) .
ʻOi aku ka nani o ke kānāwai no \(7\) , \(11\) a me \(13\) i ka manawa like. Hoʻopaʻa ia i kēlā
\[
1001 = 7 \cdot 11 \cdot 13
\]
a penei
\[
1000 \equiv -1 \pmod{7}, \qquad
1000 \equiv -1 \pmod{11}, \qquad
1000 \equiv -1 \pmod{13}.
\]
Inā ʻoe e puʻunaue i kahi helu i nā poloka ʻekolu mai ka ʻākau a i ka hema, hiki iā ʻoe ke hoʻohui a unuhi i kēia mau poloka ma ke ʻano ʻē aʻe.
\[
X = 123456789
\]
No laila, inā e noʻonoʻo kekahi
\[
789 - 456 + 123 = 456.
\]
Loaʻa i ka helu kumu ke koena like modulo \(7\) , \(11\) a me \(13\) e like me \(456\) No laila, hiki ke pani ʻia kahi helu nui loa e kahi helu liʻiliʻi loa me ka ʻole o ka hoʻololi ʻana i kona mahele ʻana e kēia mau helu ʻekolu.
He ʻano nani hoʻi ka mahele ʻana e \(37\) Mai ka wā mai
\[
999 = 27 \cdot 37
\]
pili
\[
1000 \equiv 1 \pmod{37}.
\]
Ke hiki ke puʻunaue ʻia e \(37\) hiki ke hoʻohui wale ʻia nā poloka o ʻekolu. No ka laʻana, mai
\[
99937
\]
ka huina
\[
99 + 937 = 1036.
\]
Aia ma laila
\[
1036 = 28 \cdot 37
\]
Inā hiki ke puʻunaue ʻia \(99937\) e 37, a laila hiki ke puʻunaue ʻia ʻo 99937 e \(37\) .
I ka wā mua, ʻike ʻia kēlā mau lula he mau hana helu, akā ʻo ka hopena he mau noi wale nō ia o ka manaʻo like: ke pani nei i nā mana nui o ka ʻumi me nā koena maʻalahi modulo ka helu i nīnau ʻia. Hoʻololi kēia i kahi helu decimal nui i kahi helu maʻalahi e pili ana i nā congruences. ʻO ia ke kumu i ʻoi aku ai kēlā mau lula mahele ma mua o nā hana helu helu wale nō; hōʻike lākou i ka hōʻemi ʻana i nā koena modulo \(10^k\): Ma ke ʻano nīnau, ʻike ʻia lākou he mau pahele noʻonoʻo liʻiliʻi, akā alakaʻi pololei i kahi manaʻo kupanaha i ke kumumanaʻo helu.