{"id":3855,"date":"2014-07-24T06:56:39","date_gmt":"2014-07-24T06:56:39","guid":{"rendered":"http:\/\/eltawil.org\/sciencewonders\/?p=3855"},"modified":"2014-07-24T06:56:39","modified_gmt":"2014-07-24T06:56:39","slug":"%d8%b7%d8%a7%d9%87%d8%b1-%d8%a7%d9%84%d8%ac%d9%85%d9%84","status":"publish","type":"post","link":"https:\/\/eltawil.org\/sciencewonders\/?p=3855","title":{"rendered":"\u0637\u0627\u0647\u0631 \u0627\u0644\u062c\u0645\u0644"},"content":{"rendered":"

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\u0641\u064a \u0627\u0644\u062d\u0642\u064a\u0642\u0629 \u0627\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0631\u064a\u0627\u0636\u064a\u0629 \u0627\u0644\u0644\u062a\u0649 \u062a\u0624\u062f\u064a \u0647\u0630\u0627 \u0627\u0644\u063a\u0631\u0636 \u0647\u0649 \u0639\u0645\u0644\u064a\u0629 \u0634\u062f\u064a\u062f\u0629 \u0627\u0644\u0628\u0633\u0627\u0637\u0629 \u0643\u0623\u0646\u0647\u0627 \u0644\u0639\u0628\u0629 \u0623\u0637\u0641\u0627\u0644. \u0628\u0644 \u0644\u0639\u0644 \u0628\u0639\u0636\u0646\u0627 \u0641\u0649 \u0637\u0641\u0648\u0644\u062a\u0647 \u0648\u0635\u0644 \u0627\u0644\u0649 \u0641\u0643\u0631\u0629 \u0645\u0634\u0627\u0628\u0647\u0629. \u0627\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629 \u0627\u0644\u0644\u062a\u0649 \u062a\u0633\u062a\u062e\u062f\u0645\u0647\u0627 \u0637\u0631\u064a\u0642\u0629 “\u0627\u0644\u062c\u0645\u0644” \u062a\u0639\u0631\u0641 \u0628\u0627\u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u0646\u0645\u0637\u064a \u0648\u0633\u0623\u0633\u0645\u064a\u0647\u0627 \u0627\u0646\u0627 \u0647\u0646\u0627 “\u0627\u0644\u062d\u0633\u0627\u0628 \u0628\u0627\u0644\u0628\u0627\u0642\u064a” modular arithmetic \u0648\u0641\u064a\u0647\u0627 \u0646\u062a\u062e\u064a\u0644 \u0627\u0646 \u0647\u0646\u0627\u0643 \u062d\u062f \u0627\u0642\u0635\u0649 \u0644\u0644\u0627\u0639\u062f\u0627\u062f \u0648\u0644\u0646\u062a\u0627\u0626\u062c \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629 \u0633\u0648\u0627\u0621 \u0627\u0644\u0645\u062f\u062e\u0644\u0627\u062a \u0627\u0648 \u0627\u0644\u0645\u062e\u0631\u062c\u0627\u062a. \u062a\u0645\u0627\u0645\u0627 \u0643\u0645\u0627\u0643\u064a\u0646\u0629 \u0642\u0635 \u0627\u0644\u0634\u0639\u0631 \u0627\u0644\u0644\u062a\u0649 \u0646\u0636\u0628\u0637\u0647\u0627 \u0639\u0644\u0649 \u062f\u0631\u062c\u0629 \u0645\u0639\u064a\u0646\u0629 \u0641\u062a\u0642\u0648\u0645 \u0628\u0642\u0635 \u0627\u0644\u0634\u0639\u0631 \u0648\u064a\u0643\u0648\u0646 \u0627\u0644\u0628\u0627\u0642\u0649 \u062f\u0627\u0626\u0645\u0627 \u0627\u0642\u0644 \u0645\u0646 \u062a\u0644\u0643 \u0627\u0644\u062f\u0631\u062c\u0629!<\/p>\n

\u0648\u0643\u0645\u0627 \u0646\u062d\u062a\u0627\u062c \u0627\u0648\u0644\u0627 \u0627\u0644\u0649 \u0638\u0628\u0637 \u0645\u0627\u0643\u064a\u0646\u0629 \u0642\u0635 \u0627\u0644\u0634\u0639\u0631 \u0639\u0646\u062f \u062f\u0631\u062c\u0629 \u0645\u0639\u064a\u0646\u0629 \u0642\u0628\u0644 \u0627\u0633\u062a\u062e\u062f\u0627\u0645\u0647\u0627 \u064a\u0646\u0628\u063a\u0649 \u0627\u0646 \u0646\u062e\u062a\u0627\u0631 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0627\u0642\u0635\u0649 \u0627\u0644\u0644\u0630\u0649 \u0644\u0627 \u062a\u062a\u062e\u0637\u0627\u0647 \u0628\u0627\u0642\u0649 \u0627\u0644\u0623\u0639\u062f\u0627\u062f. \u0641\u062f\u0639\u0648\u0646\u0627 \u0646\u0641\u062a\u0631\u0636 \u0645\u062b\u0644\u0627 \u0627\u0646\u0647 \u0647\u0630\u0627 \u0627\u0644\u0639\u062f\u062f \u0647\u0648 13 -\u0641\u0649 \u0637\u0631\u064a\u0642\u0629 \u0627\u0644\u062c\u0645\u0644 \u0647\u0648 \u062f\u0627\u0626\u0645\u0627 \u0639\u062f\u062f \u0627\u0648\u0644\u0649-. \u062f\u0639\u0648\u0646\u0627 \u0627\u0644\u0623\u0646 \u0646\u0631\u0649 \u0627\u0644\u0639\u0645\u0644\u064a\u0627\u062a \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629 \u0645\u062b\u0644 \u0627\u0644\u062c\u0645\u0639 \u0648\u0627\u0644\u0637\u0631\u062d \u0648\u0627\u0644\u0636\u0631\u0628 \u0648 \u0627\u0644\u0642\u0633\u0645\u0629 \u0641\u064a \u0634\u0643\u0644\u0647\u0627 \u0627\u0644\u062c\u062f\u064a\u062f. \u0648\u062f\u0639\u0648\u0646\u0627 \u0646\u0644\u062e\u0635 \u0627\u0644\u0627\u0645\u0631 \u0641\u0649 \u0627\u0644\u0642\u0627\u0639\u062f\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629: \u0646\u062d\u0633\u0628 \u0627\u0648\u0644\u0627 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0645\u0648\u062c\u0648\u062f\u0629 \u0627\u0645\u0627\u0645\u0646\u0627 \u0628\u0627\u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u0639\u0627\u062f\u064a \u0641\u0627\u0630\u0627 \u0643\u0627\u0646\u062a \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0644\u064a\u0633\u062a \u0627\u0642\u0644 \u0645\u0646 13 \u0641\u064a\u062c\u0628 \u0627\u0646 \u0646\u0642\u0633\u0645 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0639\u0644\u0649 13 \u0648\u064a\u0643\u0648\u0646 \u0628\u0627\u0642\u0649 \u0627\u0644\u0642\u0633\u0645\u0629 \u0647\u0648 \u0646\u0627\u062a\u062c \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629 \u0627\u0644\u0646\u0647\u0627\u0626\u064a.<\/p>\n

\u0644\u0646\u0631\u064a \u0645\u062b\u0644\u0627 \u062c\u0645\u0639 9 \u0648 6 . \u062d\u064a\u062b \u0627\u0646 \u0645\u062c\u0645\u0648\u0639\u0647\u0645\u0627 \u0627\u0644\u0645\u0623\u0644\u0648\u0641 \u064a\u0639\u0637\u0649 15 . \u0648\u062d\u064a\u062b \u0627\u0646 15 \u0644\u064a\u0633\u062a \u0627\u0642\u0644 \u0645\u0646 13 . \u0627\u0630\u0646 \u0646\u0642\u0633\u0645 15 \u0639\u0644\u0649 13 \u0641\u0646\u062d\u0635\u0644 \u0639\u0644\u0649 1 \u0648 \u0627\u0644\u0628\u0627\u0642\u0649 \u0647\u0648 2 . \u0627\u0630\u0646 \u062a\u0643\u0648\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0646\u0647\u0627\u0626\u064a\u0629 2 \u0627\u0648 \u0627\u0646:
\n9 + 6 =2 (mod 13)
\n\u0648\u0628\u0645\u0627 \u0627\u0646 \u0627\u0644\u0637\u0631\u062d \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0644\u0644\u062c\u0645\u0639 \u0641\u064a\u0643\u0648\u0646 2 \u0646\u0627\u0642\u0635 6 \u064a\u0633\u0627\u0648\u0649 9
\n2 – 6 = 9 (mod 13)
\n\u0648\u0627\u0644\u0645\u062b\u0644 \u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0636\u0631\u0628. \u0645\u062b\u0644\u0627 \u0627\u0630\u0627 \u0627\u0631\u062f\u0646\u0627 \u0636\u0631\u0628 5 \u0641\u0649 9 . \u0648\u062d\u064a\u062b \u0627\u0646 \u0627\u0644\u0636\u0631\u0628 \u0627\u0644\u0639\u0627\u062f\u064a \u064a\u0639\u0637\u0649 45 . \u0627\u0630\u0646 \u0646\u0642\u0633\u0645 45 \u0639\u0644\u0649 13 \u0641\u062a\u0643\u0648\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 3 \u0648 \u0627\u0644\u0628\u0627\u0642\u0649 6
\n5 * 9 = 6 (mod 13)
\n\u0648\u0628\u0645\u0627 \u0627\u0646 \u0627\u0644\u0642\u0633\u0645\u0629 \u0639\u0643\u0633 \u0627\u0644\u0636\u0631\u0628 \u0627\u0630\u0646 \u062a\u0643\u0648\u0646 \u0642\u0633\u0645\u0629 6 \u0639\u0644\u0649 5 \u062a\u0639\u0637\u0649 9
\n6 \/ 5 = 9 (mod 13)<\/p>\n

\u0644\u0643\u0646 \u0645\u0627\u0647\u0649 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u062d\u0633\u0627\u0628\u064a\u0629 \u0627\u0644\u0644\u0649 \u0633\u0646\u0633\u062a\u062e\u062f\u0645\u0647\u0627 \u061f \u0647\u0644 \u0647\u0648 \u0627\u0644\u062c\u0645\u0639 \u0628\u0627\u0644\u0641\u0627\u0631\u0642 \u0627\u0648 \u0627\u0644\u0636\u0631\u0628 \u0628\u0627\u0644\u0641\u0627\u0631\u0642\u061f \u0627\u0644\u0627\u062c\u0627\u0628\u0629 \u0644\u0627 \u0647\u0630\u0627 \u0648\u0644\u0627 \u0630\u0627\u0643 \u0648\u0644\u0643\u0646\u0647 \u062d\u0633\u0627\u0628 \u0627\u0644\u0623\u0633 \u0628\u0627\u0644\u0641\u0627\u0631\u0642! \u0648\u0647\u0649 \u0639\u0645\u0644\u064a\u0629 \u062a\u062a\u0645 \u0627\u064a\u0636\u0627 \u0637\u0628\u0642 \u0627\u0644\u0645\u0646\u0637\u0642 \u0627\u0644\u0633\u0627\u0628\u0642. \u062f\u0639\u0648\u0646\u0627 \u0646\u062d\u0633\u0628 \u0645\u062b\u0644\u0627 2 \u0623\u0633 4 . \u062d\u064a\u062b \u0627\u0646 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u062a\u0642\u0644\u064a\u062f\u064a\u0629 \u062a\u0639\u0637\u064a 16 . \u0627\u0630\u0646 \u0646\u0642\u0633\u0645 16 \u0639\u0644\u0649 13 \u0641\u062a\u0643\u0648\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 1 \u0648 \u0627\u0644\u0628\u0627\u0642\u0649 3. \u0627\u0648 \u0627\u0646
\n2^4 = 3 (mod 13)
\n\u0627\u0645\u0627 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0639\u0643\u0633\u064a\u0629 \u0648\u0647\u0649 \u0639\u0645\u0644\u064a\u0629 \u062d\u0633\u0627\u0628 \u0627\u0644\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645 \u0641\u0644\u0627 \u062a\u0648\u062c\u062f \u0637\u0631\u064a\u0642\u0629 \u0633\u0647\u0644\u0629 \u0644\u0627\u064a\u062c\u0627\u062f\u0647\u0627 \u0648\u0644\u0630\u0644\u0643 \u0646\u0633\u062a\u063a\u0644 \u0647\u0630\u0627 \u0627\u0644\u0623\u0645\u0631 \u0641\u0649 \u0627\u0644\u062a\u0634\u0641\u064a\u0631 \u0628\u0637\u0631\u064a\u0642\u0629 “\u0627\u0644\u062c\u0645\u0644”. \u0648\u0644\u062a\u0648\u0636\u064a\u062d \u0647\u0630\u0627 \u0627\u0644\u0623\u0645\u0631 \u0646\u0631\u0649 \u0641\u0649 \u0627\u0644\u0631\u0633\u0645 \u0627\u0644\u062a\u0627\u0644\u0649 \u062c\u062f\u0648\u0644\u0627 \u0644\u062d\u0633\u0627\u0628 \u0627\u0644\u0623\u0633 \u0644\u0644\u0639\u062f\u062f 2 \u0644\u0643\u0646 \u0647\u0646\u0627 \u0627\u0644\u062d\u062f \u0627\u0644\u0627\u0642\u0635\u0649 \u0644\u064a\u0633 13 \u0628\u0644 59 . \u0648\u0641\u0649 \u0627\u0644\u062c\u062f\u0648\u0644 \u0646\u0631\u0649 \u062d\u0633\u0627\u0628 \u0627\u0644\u0627\u0633 \u0644\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0632\u0648\u062c\u064a\u0629 \u0641\u0642\u0637. \u0641\u0645\u0627\u0630\u0627 \u0646\u0644\u0627\u062d\u0638\u061f<\/p>\n

\"\u062d\u0633\u0627\u0628<\/a>

\u062d\u0633\u0627\u0628 \u0623\u0633 2<\/p><\/div>\n

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\u0646\u0639\u0648\u062f \u0644\u062a\u0634\u0641\u064a\u0631 \u0627\u0644\u0639\u062f\u062f 15 . \u0648\u0645\u0631\u0629 \u0627\u062e\u0631\u0649 \u0644\u0627\u0628\u062f \u0627\u0646 \u0646\u062e\u062a\u0627\u0631 \u0645\u0641\u062a\u0627\u062d \u0639\u0627\u0645 \u0648\u0623\u062e\u0631 \u062e\u0627\u0635. \u062f\u0639\u0646\u0627 \u0646\u0631\u0645\u0632 \u0644\u0644\u0645\u0641\u062a\u0627\u062d \u0627\u0644\u062e\u0627\u0635 \u0628 X \u0648\u0642\u064a\u0645\u062a\u0647 \u062a\u062a\u0631\u0648\u0627\u062d \u0628\u064a\u0646 0 \u0648 58 . \u0644\u0643\u0646 \u0645\u0646 \u0627\u0644\u0628\u062f\u064a\u0647\u064a \u0627\u0646 \u0646\u0628\u062a\u0639\u062f \u0639\u0646 \u0627\u062e\u062a\u064a\u0627\u0631 0 \u0623\u0648 1 \u0644\u0623\u0646\u0647\u0627 \u0627\u0639\u062f\u0627\u062f \u0633\u0647\u0644\u0629 \u0627\u0644\u0623\u062e\u062a\u0631\u0627\u0642. \u0628\u064a\u0646\u0645\u0627 \u0627\u0644\u0645\u0641\u062a\u0627\u062d \u0627\u0644\u0639\u0627\u0645 \u0647\u0646\u0627 \u064a\u062a\u0643\u0648\u0646 \u0645\u0646 3 \u0627\u0639\u062f\u0627\u062f. \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0623\u0648\u0644 \u0647\u0648 59 \u0627\u0648 \u0647\u0648 \u0630\u0644\u0643 \u0627\u0644\u062d\u062f \u0627\u0644\u0627\u0642\u0635\u0649. \u0648\u062f\u0639\u0646\u0627 \u0646\u062e\u062a\u0627\u0631 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u062b\u0627\u0646\u0649 \u0645\u0648\u0644\u062f generator \u0645\u0645\u0643\u0646 \u0644\u0645\u062c\u0645\u0648\u0639\u0629 \u0628\u0627\u0642\u064a 59 \u0623\u0648 2. \u0648\u062f\u0639\u0646\u0627 \u0646\u062e\u062a\u0627\u0631 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u062b\u0627\u0644\u062b \u0628\u062d\u064a\u062b \u064a\u0643\u0648\u0646 2 \u0623\u0633 \u0627\u0644\u0645\u0641\u062a\u0627\u062d \u0627\u0644\u062e\u0627\u0635 \u0623\u0648\u00a0 X \u064a\u0633\u0627\u0648\u064a \u0647\u0630\u0627 \u0627\u0644\u0631\u0642\u0645 \u0628\u062d\u0633\u0627\u0628 \u0628\u0627\u0642\u0649 59. \u0648\u0644\u0646\u0641\u0631\u0636 \u0627\u0646 \u0647\u0630\u0627 \u0627\u0644\u0639\u062f\u062f \u0647\u0648 42.<\/p>\n

\u0627\u0630\u0646 \u0645\u0641\u062a\u0627\u062d\u0649 \u0627\u0644\u0639\u0627\u0645 \u064a\u062a\u0643\u0648\u0646 \u0645\u0646 \u0627\u0644\u0623\u0639\u062f\u0627\u062f: 2 \u064842 \u0648 59 . \u0648\u0644\u062a\u0634\u0641\u064a\u0631 \u0627\u0644\u0631\u0633\u0627\u0644\u0629 \u0627\u062e\u062a\u0631 \u0627\u0649 \u0639\u062f\u062f \u0639\u0634\u0648\u0627\u0626\u0649 \u064a\u0642\u0639 \u0628\u064a\u0646 \u0635\u0641\u0631 \u0648 58. \u0648\u0645\u0631\u0629 \u0627\u062e\u0631\u0649 \u0645\u0646 \u0627\u0644\u0628\u062f\u064a\u0647\u064a \u0627\u0646 \u0646\u0628\u062a\u0639\u062f \u0639\u0646 \u0627\u062e\u062a\u064a\u0627\u0631 0 \u0623\u0648 1 . \u0648\u0644\u0646\u0641\u0631\u0636 \u0627\u0646 \u0627\u062e\u062a\u064a\u0627\u0631\u0643 \u0642\u062f \u0648\u0642\u0639 \u0639\u0644\u0649 9 . \u0627\u062d\u0633\u0628 \u0627\u0644\u0623\u0646 2 \u0627\u0633 9 \u0628\u062d\u0633\u0627\u0628 \u0628\u0627\u0642\u064a 59 .\u062b\u0645 \u0627\u062d\u0633\u0628 42 \u0623\u0633 9 \u0628\u062d\u0633\u0627\u0628 \u0628\u0627\u0642\u0649 59 \u062b\u0645 \u0627\u0636\u0631\u0628 \u0627\u0644\u0646\u0627\u062a\u062c \u0641\u0649 15 \u0628\u062d\u0633\u0627\u0628 \u0628\u0627\u0642 59 \u0644\u0646\u0635\u0644 \u0627\u0644\u0649 \u0627\u0644\u0631\u0633\u0627\u0644\u0629 \u0627\u0644\u0645\u0634\u0641\u0631\u0629.
\n2^9 = 40 (mod 59)
\n(42 ^9) * 15 = 38 (mod 59)<\/p>\n

\u0627\u0630\u0646 \u0627\u0644\u0631\u0633\u0627\u0644\u0629 \u0627\u0644\u0644\u062a\u0649 \u0633\u062a\u0631\u0633\u0644\u0647\u0627 \u0644\u0647\u0649 \u0633\u062a\u062d\u062a\u0648\u064a \u0639\u0644\u0649 \u0627\u0644\u0639\u062f\u062f\u064a\u0646 38 \u0648 40 . \u062d\u064a\u062b 38 \u062a\u0634\u064a\u0631 \u0627\u0644\u0649 \u0627\u0644\u0631\u0633\u0627\u0644\u0629 \u0627\u0644\u0645\u0634\u0641\u0631\u0629 \u0648 \u0627\u0644\u0639\u062f\u062f 40 \u0647\u0648 \u0639\u062f\u062f \u0645\u0633\u0627\u0639\u062f. \u0648\u0644\u0643\u064a \u0627\u0641\u0643 \u0627\u0644\u0634\u0641\u0631\u0629 \u0639\u0644\u0649 \u0627\u0646 \u0627\u062d\u0633\u0628 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0645\u0633\u0627\u0639\u062f \u0623\u0648 40 \u0623\u0633 \u0627\u0644\u0645\u0641\u062a\u0627\u062d \u0627\u0644\u062e\u0627\u0635. \u062b\u0645 \u0627\u0642\u0633\u0645 38 \u0639\u0644\u0649 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0633\u0627\u0628\u0642\u0629 \u0644\u0627\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0631\u0633\u0627\u0644\u0629 \u0627\u0644\u0623\u0635\u0644\u064a\u0629. \u0648\u0627\u0646\u0627 \u0641\u0649 \u0647\u0630\u0627 \u0627\u0633\u062a\u063a\u0644 \u062e\u0627\u0635\u064a\u0629 \u0645\u0646 \u062e\u0648\u0627\u0635 \u0627\u0644\u062d\u0633\u0627\u0628 \u0628\u0627\u0644\u0628\u0627\u0642\u0649 \u0633\u0627\u0633\u0631\u062f\u0647\u0627 \u0647\u0646\u0627 \u062f\u0648\u0646 \u0628\u0631\u0647\u0627\u0646:
\na^x^s = a^s^x
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