MGM 502 Number Theory [Teori Nombor]

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1 UNIVERSITI SAINS MALAYSIA Second Semester Examination 009/010 Academic Session Aril/May 010 MGM 50 Number Theory [Teori Nombor] Duration : 3 hours [Masa : 3 jam] Please check that this examination aer consists of FIVE ages of rinted material before you begin the examination. [Sila astikan bahawa kertas eeriksaan ini mengandungi LIMA muka surat yang bercetak sebelum anda memulakan eeriksaan ini.] Instructions: [Arahan: Answer all three [3] questions. Jawab semua tiga [3] soalan.] In the event of any discreancies, the English version shall be used. [Sekiranya terdaat sebarang ercanggahan ada soalan eeriksaan, versi Bahasa Inggeris hendaklah diguna akai]. /-

2 [MGM 50] 1. (a) Use the Euclidean Algorithm to obtain integers x and y satisfying gcd(119,7) 119 x 7 y (b) Consider the roblem of urchasing RM510 ostal orders using RM30 and RM50 ostal orders. How many of each tye of ostal order should be used? Find all the ossible combinations. Define a multilicative function. Show that the function () n n is a multilicative function. Find (650) and (650). (e) If 1a rime, show that 1 1 is a erfect number.. (a) Define a Mersenne number. If is an odd rime, then any divisor of the Mersenne number, M 1is of the form k 1, where k is a ositive integer. Using this statement show that M 9 is a comosite number. (b) Using the Lucas- Lehmer Test, determine whether the Mersenne number M 11 is a Mersenne rime. Given the linear congruence 34x 60(mod98), determine the number of solutions for this congruence. Give reasons. Next, find all solutions of the given linear congruence. Solve the following simultaneous linear congruences, x 5(mod6); x 4(mod11); x 3(mod17) (e) Given mod Show that a b n and c d mod n. a c b d mod n and ac bd mod n....3/-

3 3 [MGM 50] 1. (a) Dengan menggunakan algoritma Euclidean, daatkan integer x dan y yang memenuhi gcd(119,7) 119 x 7 y. (b) Pertimbangkan satu situasi yang memerlukan embelian RM510 kiriman wang menggunakan kiriman wang RM30 dan RM50. Beraakah jumlah setia kiriman wang yang digunakan. Cari semua kombinasi () n Takrifkan fungsi multilikatif. Tunjukkan bahawa fungsi adalah satu n fungsi yang multilikatif. Cari (650) dan (650). (e) Jika 1nombor erdana, tunjukkan bahawa semurna. 1 1 adalah nombor. (a) Takrifkan nombor Mersenne. Jika adalah satu nombor erdana, maka sebarang embahagi bagi nombor mersenne, M 1 adalah dalam bentuk k 1, di mana k adalah satu nombor integer ositif. Dengan menggunakan kenyataan ini, tunjukkan bahawa M 9 adalah satu nombor komosit. (b) Menggunakan ujian Lucas-Lehmer. Tentukan sama ada nombor Mersenne M 11 adalah nombor erdana Mersenne. Diberikan kongruen 34x 60(mod98), tentukan bilangan enyelesaian bagi kongruen ini. Berikan alasan anda. Seterusnya daatkan semua enyelesaian bagi kongruen linear ini. Selesaikan kongruen linear serentak berikut, x 5(mod6); x 4(mod11); x 3(mod17) (e) Diberikan mod Tunjukkan bahawa a b n dan c d mod n. a c b d mod n dan ac bd mod n....4/-

4 4 [MGM 50] 3. (a) Solve the following quadratic congruence. 5x 6x 1 0(mod3) (b) Define a Legendre symbol. If a b(mod ) then show that a, b are integers that are relatively rime to the odd rime. a b,where (e) Using the concet of Legendre symbol, determine whether or not the congruence x 46 mod17 is solvable. Encryt the message DO NOT STOP using the affine transformation C P 5(mod6). What is the cihertext that is roduced when RSA encrytion with key ( en, ) (7,67) is used to encryt the message LIFE IS A DREAM....5/-

5 5 [MGM 50] 3. (a) Selesaikan kongruen kuadratik berikut 5x 6x 1 0(mod3) (b) Takrifkan simbol Legendre. Jika a b(mod ) a b maka tunjukkan bahawa, di mana a, b adalah integer yang secara relatif adalah erdana terhada nombor erdana yang ganjil. (e) Dengan menggunakan konse simbol Legendre, tentukan sama ada kongruen x 46 mod17 memunyai enyelesaian. Enkrikan esanan DO NOT STOP menggunakan transformasi afin C P 5(mod6). Aakah teks saifer yang dihasilkan aabila enkrisi RSA dengan kekunci ( en, ) (7,67) digunakan untuk mengenkrisi esanan LIFE IS A F DREAM. [100 markah] - ooo O ooo -

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