Question

Problem 5

1. Find the values of (379) and (4725). 2. Prove that for any m > 2, (m) is even. 3. Prove that if (371) - 36(n) then 3|n. Hint: Try proving the contrapositive. 4. Suppose that a =b (mod m), a = b (mod n), and ged(m, n) = 1. Prove that a = b (mod mn). 5. Use Euler's Theorem and the method of successive squaring to find 56820 (mod 2444). That is, find the canonical residue it is congruent to modulo 2444. Note that the prime factorization of 2444 is 2444 = 22. 13. 47. Optional challenge problem. Prove that for all m, n e N, 6(lem(m, n)) (ged(m, n) = (m) (n).

Answer 1

UM 12 COD suppos 1. 8 (379) = 378 [since 379 is prime & (4725) = 8/3² x 5² x7) = 3 +57(1-2 (1-1)×(1-24) - 2160 suppose ny If n has an odd prime. factor, say & then napkin ged (m. P) 21 and & (n) = o(ok) Q (m) 2(P-1) pk-1 & (m. even [P- in even ] It has no odd prime factors, then naak with K7, so q (2 k) = 2h even. I 3. 9 (3n) = 30 (2) (proved, & suppose ged (3,0) 1 = Q(34) 20 (3) 9 (6) . & 3914 ) andit ei ged (3, 0) 1 2 34 n. (prored a = b (modn) & aab (modn) & a = b (modn) and and god (minal arba kim ota-bakan & Kim akan since ged (min) a1 = n/K, Emkz arba nkzm a mnkz 2) K, ankz i.aab (wood mn) (proveds.