Discrete Math Interesting Exercises - Base Conversion, RSA, Linear Congruences, Transposition Cipher

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Question

P1. Convert (7206)8 and (AOEB)16 to binary.

P2. Express gcd(84, 119) as a linear combination of 84 and 119.

P3. Use the Chinese Remainder Theorem to solve the following system of linear congruences: x ≡ 1 mod 4 x ≡ 2 mod 5 x ≡ 3 mod 7

P4. Encrypt the message GRIZZLY BEARS using blocks of five letters and the transposition cipher based on the permutation of {1,2,3,4,5} with σ(1) = 3, σ(2)=5, σ(3)=1, σ(4) = 2 and σ(5)=4. For this problem, use the letter X as many times as necessary to fill out the final block of fewer than 5 letters.

P5. A check digit a13 for an ISBN-13 with initial digits a1, a2, …, a12 is determined by the congruence {(a1 + a3 +….+ a13) + 3(a2 + a4 + … +a12)} ≡ 0 mod 10. Determine if each of the following 13‐digit numbers is a valid ISBN‐13. a. 978‐0‐073‐20679-1 b. 978‐0-201‐10179‐9

P6. Determine the plaintext for the RSA ciphertext 2169 0628 5540 2169 6560 6401 4829 with n = 7747 (p = 61 and q = 127) and e=17.

Solution Preview

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P1. Convert (7206)8 and (AOEB)16 to binary.
Answer:
We first convert the number from octal to decimal and then to binary.
(7206)8=7*8^3+2*8^2+0*8+6=7*512+2*64+6=3,584+134=3,718
Now we transform the decimal number into binary.
3,718:2=1,859 remainder 0
1,859:2= 929 remainder 1
929:2=464 remainder 1
464:2=232 remainder 0
232:2=116 remainder 0...

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