Problem 3:(a) Compute 71529 (mod 51). Show your work.(b) Compute 9-1 (mod 19) by listing the multiples. Show your work.(c) Compute 9-1 (mod 19) using Fermat's Little Theorem. Show your work.(d) Find an integer x, 0 < x < 18, that satisfies the following congruence: 9x + 13 = 10 (mod 19). Showyour work. You should not use brute force approach.

“Since you have asked multiple questions, we will solve the first question that is (a) for you. If…

Answered: Problem 3: (a) Compute 71529 (mod 51).…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 1: (a) Compute 13-¹ (mod 23) by enumerating multiples. Show your work. (b) Compute 13-¹ (mod 23) using Fermat's Little Theorem. Show your work. (c) Compute 11-11 (mod 19) using Fermat's Little Theorem. Show your work. (d) Use Fermat's Little Theorem to compute 71209643 (mod 11). Show your work. (e) Find an integer x, 0≤x≤ 40, that satisfies 31x + 42 = 4 (mod 41). Show your work. You should not use brute force approach. (f) Calculate 138-1 (mod 2784) using any method of your choice. Show your work. (g) How many integers have inverses modulo 144? Justify.
You are trying to factor n = 2288233. Suppose you discover that 8805252 = 2 , 20572022 = 3, and 6485812 = 6 (mod n). Use this information to factor n. Please give detailed steps! Really trying to understand this problem.
8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10. Apply Fermat's Little Theorem to find the remainder when 22⁰ + 330 + 440+550 + 660 is divided by 7.
(a) Compute 7^35 (mod 3) .. Please explain! (b) Find the last three digits of 9^30 (Please use Euler's Theorem AND the Chinese Reminder Theorem)
Q7 Compute 59^11 mod 13 Note: You should be able to do this using the modulo squaring method and show your work.
1. (a) question: Use Fermat’s Little Theorem to calculate:(i) 23106 (mod 107) (ii) 431 (mod 7) (iii) 316 + 532 + 864 +13128 (mod 17)(iv) the remainder of 21000 divided by 13
(10) Prove that for any prime number P>3. p² = 1 (mod 24) (Hint: chinese remainder theorem tells us SP² = 1 (mod 3) p² = 1 (mod 8) need to show we only need to p²=1 (mod 8), which we know that P PH two integers. The hard part is to show means S1 p²2-1 81 = (P-1) (PH). must be odd. Hence, P-1, So it is enough to show 2 | P-1 F are PH).
If you add n whole numbers (even or odd) when is the sum of the numbers odd? Translate into adding O's and l's (mod 2). When do they add to 1?
You are trying to factor n = 2288233. Suppose you discover that8805252 = 2 , 20572022 = 3, and 6485812 = 6 (mod n). Use this information to factor n.
4. (i) Using the fact that 187 = 11 · 17, show that 2186 = 174 mod 187. This shows that 187 fails the base 2 primality test. (ii) Compute 21911 mod 1911. Use Fermat's little theorem and hand calculations. Does 1911 pass or fail the base 2 primality test? You may use the prime factorization of 1911.

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Problem 3: (a) Compute 71529 (mod 51). Show your work. (b) Compute 9-1 (mod 19) by listing the multiples. Show your work. (c) Compute 9-1 (mod 19) using Fermat's Little Theorem. Show your work.